I can start sir, student team from IIT Madras. We can start. Ma'am can I start? Yes. Okay fine. Good morning Sir. Good morning to both. Good morning. Good morning. Amitavani SPLC. You can start the introduction. Good morning. Amitavani is the SPLC. You can start the introduction. Sankar sir, you can initiate. Dr. Sankar. Yes sir. Yes. Okay. Very good morning to all. Principal of our college, M.A. Venkatesh, principal S.I.R.T.C. Naseek, Dean R.N.D., Dr. J.B. Gurusar, I.S.I coordinator, Dr. Bharat Gorkar, today's resource person, S.I.R.T. Ma'am, senior project officer, I.T. Madras, Priyadarshan Ma'am, and my dear friends, I, Dr. Srinivas Sankar, SPLC of Amritvani College of Engineering, Sangamner, welcomes to you all. The importance of today's workshop is domain certification. How the any faculty need to get domain certification, it will be explained in this workshop. Then, if you are doing 8-week NPTEL course, then you will be getting 1-week FTP certificate and etc. Then, as considered with student-oriented courses to hire the job, what are the different kinds of job-oriented courses, it will be explained in this workshop. Next, to improve the communication skill, NPTEL is providing the soft-skill workshop. Then, finally, then what will be the criteria to become the NPTEL star that it will be going to explain in this workshop. So, all this point, it will be going to cover in this workshop. Now, with this, I would like to request to an our principal, Dr. M. A. Venkatesar, to say a few words on this application. Sir, please. Thank you, Dr. Sunder. And good morning to all. I'm very happy to join this NPTEL program, which is streaming by IIT Madras. Finally, at last, the NPTEL has taken a big leap and it has initiated and spreading across the nation. And penetrating to all the rural colleges also. So, as a rural college of engineering, Amrita Wahini, so it is established in 1983. And so far, we have more than 12,000 plus students have been graduated. We're having eight courses with the intake of 60-60 numbers with a PG of MBR 16 separately. And we have more than 50 PhD holders and another rest 60 people in workshop completion. So, in new courses, we'll be all PhD holder colleges, we can say it. At the same time, I'd like to inform, update all the audience that our college got A plus NAC, that is 3.40 in the one and a half year back. So, it is highest among the top engineering colleges. We have 330 engineering colleges in Maharashtra, among top five, one of the college. And rural is first, we can say. So, that is one of the credential for us. And we have a graded three branches of engineering and another three are applied. And we'll do course, we are going to do it. At the same time, we also received the best engineering colleges from Pune University. And recently, that is 21, 2021. And also, we are recognized as international placement for Japanese by the Technopro, one of the company from Tokyo. It is also coming for placement trial. We have more than 24 students have been placed in Japan. And we have actually acknowledged for the Japanese learning center, apart with German language also. And we had targeted the more than 100% placement. The recent last year, we have achieved 80% plus the students have been placed. And there are many accreditations, applause and acknowledgements has given various agencies here from the U.C.T., DT and University, et cetera. At the same time, I would like to happy to share all of you is that I am receiving the best engineering college principal award tomorrow from the University Foundation Day. And with this, I wish all of you, at the same time, this is a very good activity that is NPTEL. It is an enhancing learning, what do you call that? So normally, University-Silver's being only understand the subject and we write for the examination purpose. But here, actually, it is your learning that happens. When learning happens, many things happens, like innovation or product development, many, many things. You can look into the research activities, many things. We have taken a lot of initiation from the engineering college side, like getting an IPR and also giving supporting financial assistance and entrepreneurship side, et cetera. Many things have been initiated. Of course, there's not the platform to go on elaborate achievements, but I only think wish this NPTEL program more and more, it will be active and to reach to the every student, let every student and also faculty should enroll this program and get the higher knowledge. Once the higher knowledge is acquired, the confidence will be there, the confidence, they can do wonders. So this NPTEL do wonders for all of us. With this, I wish one and all, thank you very much. And also, I thank you IIT Madras for being the initiator and taking good steps to reaching most of the rural colleges. Once again, I thank you very much. Thank you to all the coordinators who are enrolled in this program. Thank you very much. Good day. So over to Sankar sir. Thank you, sir. Thank you. I would like to... Hello. Am I audible, sir? Hello. Yes, sir. You are audible, sir. You are audible. Please, please. Good morning, dear students and faculty members. Respected Dr. Venkatesh sir, principal of Amruta Eni College of Engineering Sangam, Dr. Bharti madam, Dr. Sankar sir, Borkar sir, Dr. Vedesh sir. Myself, Dr. Millan Patil, principal of Sanjeev Institute of Technology and Research Center, I welcome you all to this NPTEL e-awareness workshop. First of all, I congratulate the NPTEL team of IIT Madras for winning the prestigious Barton Kears Reimagine Education Gold Award in the category of lifelong learning. NPTEL online courses played a vital role in teaching learning process, especially during the COVID-19 pandemic situation and help the students to clear their concepts and groom their personality. Also, NPTEL courses and its mechanism help the institute in NBA and NAC accreditation process and different ranking frameworks. Talking about our local chapter, we established the local chapter under the mentorship of IIT Bombay in 2017. Right from inception, we have been nominated as active local chapter rating except year 2020 and 2021. So far, more than 200 students attended the exam and 130 students successfully completed the NPTEL online course. Some of them are top 1%, 2% and 5%, which is really proud moment for us. We conducted the NPTEL awareness workshop each year to motivate students to enroll in the courses and register for examination. We provided the financial assistance in terms of 100% registration fee viewed off to selected faculty members of our institute based on their performance in the examination. In the end, I urged the students to take maximum benefit of NPTEL courses in terms of various initiatives such as internship, get exam preparation, etc. With this, I expressed my gratitude to NPTEL team and best wishes to SIDRC local chapter in future endeavors. Thank you. Thank you very much. Hold 2, sir. Thank you, sir. Over to Priya Darshani, madam. Yes, sir. Many of you may be familiar with this logo. This is the NPTEL logo. We'll start off with it. So NPTEL is today a lot of initiatives put together. It's not just about creating courses and putting it up on the website. But we have certification, not just certification for one course, but it is also clubbing courses together to make it into a domain. For faculty, we have FDP courses which fetch you career points. Then we have internship opportunity for students. We have soft-skilled training going on for local chapter students and so on. So this is what this presentation will talk about. Okay. So you might have all heard about these two terms, NPTEL and SWYAM. So SWYAM is a project of the ministry, the government of India-MHRD. And they created this in 2017 and said that all these agencies that you see below, IAM, Bangalore, UGC, CEC, NCRT, NITR, AICP, they all have to offer courses for certification under SWYAM. So NPTEL is also a national coordinator under SWYAM. We primarily offer courses in engineering, the UG level, the bachelor's level, the MTech ME level, some PhD level courses also. But since we have courses that are there in the management and humanities division inside IITs, we offer that also on NPTEL. NPTEL also coordinates with some institutes like the ISERS to offer, say, the physics, chemistry, mathematics courses. We work with the triple ITs. We work with National Law School Bangalore. We work with some medical institutions to offer medical-related courses. So maybe 90% of our courses will be engineering-related humanities management and 10% is something else we help. So NPTEL comes under SWYAM and we coordinate this primarily. There are other agencies also. So one thing to remember, when you are going to SWYAM.gov.in and you try to enroll to any course or you try to join any course, please check whether the national coordinator is NPTEL if you want to join the NPTEL course. Because if you just search for, say, a database management systems course, right? Maybe there is some other college also offering it with somebody else's coordinating. So the course name that we offer will also be database management system. There's will also be that possibly. So how do you actually differentiate if it's an NPTEL course? On every course card, we will have a logo of NPTEL. I showed you when we come there. But you can check that and then you can join the course. That is something that you have to look out for. NPTEL started in 2003. The vision, mission that we have had is to record the content that is being taught to IIT students and put it out for anybody to watch and view so that nobody can say, I didn't get a chance to prepare for IITs. I am not in the IITs today. I cannot get access to their knowledge, to their teaching. Hence, I denied this opportunity. Nobody can say that today. More than 2000 courses we have in our repository. You can go there and you can actually access the courses for free. You can just go there, you can scroll, you can watch all the videos. We have more than 50,000 hours of video in our website. You can access all that for free. There is no charge. There is no fee at all to access our content. There's somebody who's asking, is NPTEL useful for students of LLB? So National Law School Bangalore NLSIU is offering two courses. I think in Jan, I don't know if we have in July, we had two courses, I think, and more are coming in that. So specific to law, we do have a couple of courses. IIT Kharagpur has a law department. So from there, we have some courses coming in, but otherwise you can access the general courses that we have also. Say on management or English, those might be useful to you. So whenever you think of a website, what really do you look out for? Say how many people access that website? How much content do we have? In YouTube also, whenever we go to and watch a YouTube video or a YouTube channel, one of the things we look at is how many people have watched that video or how many subscribers are there in that channel, correct? So this is the NPTEL statistics. We have more than 1.3 billion views of the NPTEL videos, okay? So this is like huge, huge for a technical channel which has no entertainment. You can't call NPTEL videos as entertainment. There is no song and dance or something. It's all faculty from the IITs, IAC teaching serious stuff there. And for that, we have more than 1.3 billion views. We have 37 lakh subscribers on our YouTube channel. So any of you has not subscribed to our channel today. If you are missing out on something, go to our YouTube channel, subscribe for our updates so that you get it. Go to our social media handles, subscribe to our social media handles so that you get the latest updates from NPTEL. And this is where we are today as of NPTEL, okay? So these are what we do at NPTEL today and if you do an NPTEL course, what is the use of it? So the very first point is credit transfer. I was just talking to Dr. Satish also about it but maybe we should try to see whether students can get credit transfer for the courses that you do from NPTEL. But yes, even if you don't get credit transfer, I think you get a lot of knowledge from NPTEL courses. For instance, say, what are the kind of courses we have that we would recommend to students in your college, correct? So we will say that we have a lot of programming courses. So we have programming and C, C++, Python, Java programming and all that. So if you are, say, a BSE Physics student or a BA English student, maybe a BCOM student, as part of your program in the college, you would not be learning programming possibly. So you can go to NPTEL, you can learn the programming course, you can learn a bit of programming and also get that certificate along with your program.  They will have some kind of coding that is there. They will have some kind of software within their company. It's good for y'all to know at least something about it, okay? That is the first part. The other thing, okay, what are the other courses we have that is useful for you? If you're planning for higher education, you want to go in for a master's degree. Supposing you want to strengthen your fundamentals, your foundations. You want to maybe do some more courses from other institutes so that you're confident when you're attending interviews. Those courses will be available on NPTEL. So we have all the UG undergraduate courses. You can go through that. You don't even have to write the exam for each of these courses. You can just go through the videos. You can try solving problems. You can watch many videos, try to get knowledge from it and strengthen your foundation. So if you're going for higher education, I would say the courses are really useful. So first thing I'll say, try something like programming courses. When you go in for higher education, probably you want to try and strengthen your fundamentals. The third thing, try to learn something that is of interest to you, which you possibly cannot learn otherwise. For instance, we have courses on German language, teaching you German, teaching you Japanese, teaching you Sanskrit. So if you want to learn any of these, you can come to NPTEL and learn. So for instance, German is taught only by maybe the Goethe Institute and maybe some other institutes. Here the same course is taught, but it is free for you. And if you want a certificate, exam but otherwise too, you can learn the German language for free. Same thing, you can learn Japanese language. Why are these useful? Maybe you go to a company that has branches in Germany, Japan. They have opportunities for people to go there, but you need to know the language. So at that point, this becomes very useful for you. So you can try to learn other languages also through NPTEL courses. Okay, maybe you're technically very strong. You have good foundations. You have good knowledge in your particular program, but maybe your English is not very good. Hence, you're not confident about speaking. We have a lot of courses like soft skills development, soft skills enhancement, how to attend an interview, the body language that you have to have when attending an interview, interpersonal communication skills, leadership skills. So we have a lot of courses like this that you can also go through. It will give you a lot of tips on how to present yourself, how to ready yourself for such situations so that having the knowledge, you can also be confident when you talk to somebody and you are presenting or being in an interview. So that's another set of NPTEL courses that we have as part of what we offer. So anybody who wants to learn, I think should be able to get at least two or three courses of their interest that they can join and learn. Joining is free, like I said. There is no fee at all. It is all completely free and you can watch it. There are no live classes. These are all recorded lectures. So you can watch at your convenience. If your college time is, say, 9 a.m. to 4 p.m. or something, in the evening you have time. You can sit and watch NPTEL lectures. On Saturday, Sunday, you can watch it. There'll be one assignment every week. You have to solve the assignment and you have to submit it. So that also you get 10 days of time. So you can watch the lectures. You can try solving the assignment. Maybe a couple of you friends are joining the same course. You can all sit together. You can learn from each other. You can solve together. Take help from your teachers and you can do the NPTEL courses. So there is no such thing as there'll be live lectures. Maybe you'll miss a few lectures because you have college or something. It is all video lectures along with assignment. So the first part is credit transfer courses. Apart from it, if you do very well, maybe we should go to next slide. So this is the credit transfer. I don't want to discuss too much about this because I think your university doesn't have credit transfer. So I'll skip this slide. So credit transfer just has a piece of information for maybe the faculty who are out here. IATs do do credit transfer from NPTEL courses. We do it at the BTEC level. We do it at the master's level for PhD working professionals for them to get credit transfer also we do it. So IATs also do credit transfer from NPTEL courses. It is not just the outside colleges that have to do ourselves do credit transfer from NPTEL courses. This is an important thing I would say. So we have a program by which the topers in the courses actually get internship opportunity with the faculty in the IATs. So whoever has offered that course they take it normally about 2, 3, 5 interns during summer or winter if you are a topper in the course. So if you do a course very well and you become a topper in the certification course then we give you an opportunity to intern with the faculty. So we are trying to scale up these numbers COVID hit us also so we could not get them but we hope it will be at least 200, 300 students who can get the IATs with the faculty. So this is a wonderful opportunity for you to come and stay in the city be in the IAT Madras or IAT Kanpur or IAT Karakpur or IAT Bombay lab. You get to interact with the faculty members there you interact with their PhD scholars there you can do all of this which will be wonderful if you are especially thinking about going for higher studies. You will get the idea of what is done in these institutes. I think it is a very good opportunity for students outside of the IATs to come in and experience the IATs here. So there is internship then we also do soft skill training so there are many colleges who provide it on their own but there are also some colleges who do not have this training for the students. So if college students in local chapters they participate in NPTEL courses then for those students we are providing soft skill training. So this training is of two types we do a one week training or we do a two week training depending on how much time the student can spare and the college also wants the students to do. So here we have group discussion practice we ask you to write your resume and we give input on the resume we ask you all to present about yourselves give feedback about how you should talk where you should look your body language and so on and there are also a one on one like every student will get half an hour with a trainer to discuss about how they should also do. So the right side photo you see is a one on one where we do mock interviews. We try to simulate an interview and see how you're responding and then give feedback on how you should do better in your interview. So we do soft skill training for students who do internship courses. So this is something that we do from here. If you look at numbers we have finished soft skill training for more than 2000 students currently and we are doing about 10 colleges a month and we hope to scale it up. So if you're I don't know whether college has done it but yes we can extend it to you also and we can get it done. Okay, so there's a 7 of 14 day training program that I was talking about next. Okay so somebody was asking will we get jobs with these NPTEL courses how is it helpful to us and so on when we do surveys of NPTEL students and where they have gotten placed these are the companies that they have gotten placed so far. This is like some of the companies it's not all the companies that they've gotten placed for but they have said that the NPTEL courses have really helped them when they went for the interviews. When they show a certificate to the interviewer and say I did a course with the IITs I got so much marks in that course it really does right in the job interview. So those people also ask questions on that assuming you've studied well and you can answer it will definitely help you that outside of the college syllabus you have done something more to learn and skill yourself. Also companies approach us and ask us do we have students in particular areas that we can send them their resumes so we also collect resumes of students on our site and if they're interested we send these out to the companies call them for interviews go through their levels of recruitment and we have been able to place NPTEL students from our side also in some companies so we are still working on it I hope this will increase some companies on this front but otherwise even if you have like two or three courses from NPTEL and certificates for it I think it will be hugely useful because of the logos we put on the certificate okay. The next one there are a lot of students, learners, faculty who do multiple courses with us there are people who do the courses maybe you know they do three courses in a semester four courses in a semester they are toppers into a three courses every semester there are people who have done a lot of courses with us I'll maybe show you the data on the next one but this is another type of recognition we give to NPTEL learners so you can go to NPTEL website and at the top we have a link called NPTEL stars so click on that link click on any of this you will see how many people are actually doing multiple courses with us so if students are thinking oh I don't have time and I am so busy I can't do it you should just see these pages to see how come these people who are working professionals students in colleges just like you or who are maybe you know studying or faculty or something how are they able to get the time to do so many courses if they can get the time I think anybody can get the time right you just have to manage your time on when you are going to study and not waste time on something else that is not important etc so this page I would say please go and see so these are the kind of domain scholars superstars evangelists and all we have we have more than about 3000 as I can just calculate on the flight that we are going to have these people doing multiple courses and they are NPTEL stars with us today so there is something called domain certification that we have currently come up with in the last two years so what is a domain right so a domain we have created just to club courses together and maybe about five or six courses we have put together and we say that if you study this six courses and you get more than 60 in the course in each course not just pass it by getting a 40 then you get a domain scholar certificate so why is this important this is important because it shows that you have specialized in a particular area so for instance we have a programming domain so in the programming domain we have courses such as c c++ python database management course and then maybe cloud computing course and so on so if you do a programming domain right and you do all the six courses it shows that you have a solid foundation in this particular area so when you go for an interview and you actually show a domain certificate versus just one or two certificates from us that will make more of an impact on the interviewing person so this is something we recommend it is more just to guide students on what to study to become to become and specialize in one particular area we found that many NPTEL learners do four or five courses but they do random courses that are not connected to each other so this domain will guide you in what to study if you want to specialize in a particular area we have about 53 domains today in all branches of engineering management English and for faculty we have a faculty domain basic that part of it so you can go to our website there are more details on domains there you can we can talk about it later too okay yeah so these are some of the domains we have so for you it might be relevant the English studies we have a domain psychology we have a domain management we have it in marketing operations patents and intellectual property in economics and of course engineering and all we have a couple of domains so if you are a general student not an engineering student I would recommend the programming domain for you if you want to try out something do the programming domain or do the management domain that will be really useful for you so if you are an engineering student of course you can try any of the domains relevant in your particular discipline of engineering or you can do a cross discipline so for instance I am maybe mechanical engineer I like electrical engineering you can actually do the courses in electrical engineering and you can get a domain from there so here nobody stops you and says hey you are in mechanical engineering department you cannot study electrical engineering say power systems course power electronics course you are not allowed to study that there is no such thing like that anybody can study any course as long as you can cope with it so you can also do a cross department study by doing domains from any of what is listed here this is what a domain certificate looks like we give this we list all the courses you have done and we assign it out of course from the specific IIT that we are giving it from so how do we do lab courses is it online not really we do it in the institutes we do it in various institutes such as ICERS IIT you can come here you can stay for a week at the institute you can do the lab experiments you can listen to faculty giving one or two lectures in the day and then do a lab exam and take a certificate so what is the advantage of this you will get to see really modern labs you will get to see what the state of the art labs are in these institutes you can see the equipment that is there the modern equipment that is there you can see the advanced research that is happening here you can interact and get explained on all this equipment so we had very successful labs being run at all these places this is a wonderful opportunity again if you do the theory course in NPTEL those people we will permit that you can apply and come for these lab certification courses and you can do the workshops here okay so that was the idea behind the question we asked you what else do we do at NPTEL this is for faculty in colleges faculty in colleges under the AICT scheme we explicitly have an MOU with them where NPTEL advanced courses if you do and pass you will get an AICT approved FTP certificate so that is for faculty in AICT colleges for those under the UGC scheme they have given a broad guideline that if you do and Swayam courses there will be some career progression points but the actual logistics I am not sure you will have to check within your sorry university so that is something that is there but we see that more than about 20-30 thousand faculty that are exams every semester and most of them apply for the FTP certificate also this is very very useful for faculty I will say because you don't have to spend money you don't have to take leave you don't have to miss your college work and go to some institute stay for a week or stay for 10 days and then do an FTP course you can do the FTP course online itself just like the students you join for free on Swayam portal you go through the course submit assignments and then you pay the thousand rupees and attend the final exam and get your certificate once you get your certificate you can apply for the FTP certificate to us and get that too that's all it takes to do an FTP course with us today so this is very important for faculty I hope this is clear this is what the FTP certificate looks like you can see that it has the signature of the director from AICT also there is a regulation under which they have approved this the AICT logo is there so this you can present when you are going in for your promotion interviews the conversion that they do is a four week NPTEL course is equivalent to half FTP of one week an eight week NPTEL course is a full FTP of one week and a 12 week course is equivalent to one and a half weeks FTP equivalent so this is the equivalent of it yes the next thing I wanted to talk about was gate coaching so we have started a new link on our website and somebody has given us CSR money for developing this and providing whatever I wouldn't call it coaching but yes gate preparation whatever we can help students with to have and prepare for gate exams free again we are trying to do from our side so you can go here and we have given solutions to all the previous gate questions we have given videos solving those questions we have extra questions that are coming up practice questions that will be coming up small courses linking all the videos that are relevant to gates so we will be coming up phase one is only over phase two and phase three we will be coming up within the next six months so keep watching this link on NPTEL website if you are preparing for the gate exam that is one new initiative we have come up with the next initiative exams outside of India so till 2020 we were doing exams only in India where you have to go to a centre and you have to write it and then we got a lot of requests from people outside of India saying they also study from NPTEL courses and they want to write exams with us so now we help to do that also people outside India can also write exams NPTEL exams and get certificates from us okay yeah so translation is the next effort so for people who are asking in what languages do we offer courses the faculty teach only in English for now but the English transcripts we translate into eight languages so we do Tamil Telugu, Malayalam, Kannada Hindi, Gujarati Bengali and Marathi so that's the languages that we are translating it into so Telugu we have a lot of transcripts that are translated into Telugu language so the faculty will talk in English but you can also read what we have translated in Telugu or Tamil or Hindi whatever is your language and you can understand it better we are keeping the videos in English so that you also learn English because when you go to work somewhere English will be the primary language you have to communicate in so this is something we are doing one or two courses we are getting in regional language for instance this time in January we are doing a machine learning course in Tamil language the faculty is going to teach in Tamil language there is a signals and systems course that is coming out in Hindi language in July semester we are getting an R programming course in Hindi language so only two or three we have most are in English we don't have too many other language courses so that's something you can check let me take a question from the chat there are one or two saying I am from commerce background I am from BCOM what courses we can take so for BCOM students we do have a lot of accounts related courses you can check those out go to management department I will show you how to filter from it you can go there and you can see how to do it so you can check out management related courses in which we have accounts related commerce economics related courses that's something you can learn or if you are learning all that anyway in your college I would strongly recommend programming courses for you all learn at least one programming language to know what is programming how do people program whether you write the exam or not I would say please write the exam but yes even write the exam at least try a course do the assignment see whether you can do something about it or if you are not doing that I would propose the English courses I would propose the you know the German language or the Japanese language course that you can learn from those are what we have quite a few for you to choose from I would say more than 100 we would have for BCOM level students at least okay yeah so these are translation how many courses we have in each language so we have 48 courses that are translated for now so that's what we have right okay we have also something called special lecture series so we also have industry people come in and present lectures so these are again free to watch very interesting topics very very distinguished people who come and present lectures these are vice president CEO CTOs of companies like IBM cab Germany in forces that we've had in the past IAS officers who have come and spoken to us film directors who's come and spoken to us journalist in the Hindu a director at Hindu entrepreneurs so I would really say you should go and watch these lectures when you have time even if you missed it last year it's okay but listening to such people is very inspirational you know what they have done how they have worked how they have brought up new companies how they started a company where did they get their idea these are all really interesting to listen to I would say please go back and listen to some of these lectures it will give you a tremendous amount of information knowledge and maybe there'll be some topic to talk about also when somebody asks you something it will widen your knowledge base okay this is something that you can see and if you're wondering saying how many students study how many courses right maybe you think some students study only one or two courses and that's all we can do that's what is useful right look at this there is one learner who has done forty seven NPTEL exams with us that's as good as a degree program or more than the degree program that all of you are doing how many courses you'll do there I don't know maybe if you're doing a BSE it's six into six maybe thirty six courses or if it's a B take also if I say six courses per semester it's about forty eight so you're equivalent to that okay so there is one learner who's done forty seven exams thirty four exams thirty three exams thirty two exams and so on so you can see maximum people yes 5.6 lakhs do one exam but there are 1.4 lakhs who are doing two exams forty eight thousand doing three and if you look up to ten exams even right lot of them are doing ten exams eight exams with us so maybe you should think about and plan for how you're going to study which courses you're going to do as you are in college and every semester try to do at least one or two NPTEL courses and finish with a certificate that will be really nice that you can do that with your college okay this is something we wanted to say who are these people who are doing these courses they are employed they are faculty members they are students all people like you so it's not like these people come suddenly from somewhere or they are also special they are so privileged nothing like that they are there in all towns in all kinds of environment they also have their constraints but they are also studying from these courses and doing multiple courses okay this is what the certificate looks like it has the svayam logo and it has the okay it has the NPTEL logo then it has your photo on the certificate it has as you can see the elite tag the gold medal we have just put it here to say if you fall under these categories you can see all these coming on your certificate then the institutes if IT Madras offers the course at the bottom where the institute logo and round this foot you can see IT Madras logo there IT Madras the name will be put there signatures from our side will come and so on so this is the way that you can see how the certificate looks we had 20 lakh people who joined our courses in January semester 20 lakhs who joined our July semester and in 2021 we had 16 lakhs who joined the Jan semester 16 lakhs who joined our July semester so it's not 5000 it's not 10,000 it's not 1 lakh like I'm seeing on the chat window here but it is 16 lakhs who was our enrollment number last semester out of these exams those who have written over the last 5 years it is 15.1 lakh people have registered for exams with us and they've gotten so our total enrollment in NPTEL today is 1.58 crores over the last 7 years from when we are running this certification project we have finished 3500 courses almost and enrollment is 1.58 crores 15 lakhs exam registration that's the thing so you can see how big NPTEL is NPTEL is not something that we are doing 100, 200, 500 people or something like that every semester when we conduct exams in March and April it's 3 lakh people who write our exams so that's the level at which we do we conduct exams all over the country in about 170 cities we conduct exams outside India like we said so this is our data so if somebody tells you okay I haven't heard of NPTEL you should really tell them saying that 15 lakh people enroll every semester and 3 lakh people write exams you're missing out you know that's the answer I will give that okay you can see a lot of feedback about NPTEL on our social media handles we keep regularly posting about it we are present on Instagram, on Facebook on YouTube community and you can see all the feedback that we have posted there about people who have gotten jobs people who have gotten through the soft skill training who have done our courses, course feedback everything is available there so do subscribe to our social media handles so that you can get news from our site okay I'll just give a brief on NPTEL local chapter because I don't know how many of you are faculty from different colleges who are not a local chapter with us so NPTEL works with about 4,500 colleges today across the country and we have coordinators in each of these colleges who are called a single point of contact or SPOCs and through them is how we are so successful so thanks to all the local chapter SPOCs whoever is watching this workshop today was not watching the workshop today whoever is anywhere they are the champions for NPTEL within the colleges without them I don't think NPTEL would have grown so much so thanks to all the SPOCs of the local chapters and I would like to just briefly say how you can become a local chapter if there's somebody watching who's not at a local chapter so to become an NPTEL local chapter there is no money involved there is no financial dealing you just need to send an email saying that you want to become a local chapter there's a google form and the website fill it up and we need a coordinator in your college who will be our local champion so the head of the institution the director the principal whoever has to send an email saying that we would like to become a local chapter and so and so will be the single point of contact for NPTEL and that's all once we receive the email we'll verify some details and we will make you a local chapter so what why should you become a local chapter can students and faculty from colleges who are not local chapters participate in NPTEL yes totally you can participate you can be a free learner you can learn from our courses without belonging to a local chapter but having a local chapter in a college is a big advantage there are logins that the single point of contact to the SPOC has one is a login on svayam portal one is a login on our NPTEL website and here they get information related to enrollment exam registration the marks the transcripts the certificates all of this is given to the SPOC directly that they can actually see and they can share with their management also the other benefits are if we don't have a center in your particular city you can request for that if they have more than 200 people writing exams we can facilitate centers in your particular city faculty can become mentors you can follow the progress of your students in the colleges that you can do another important thing I will say is we have CSR support from companies who also give us money to help students from economically weaker backgrounds to write exams so this fee waiver we give only to local chapters so if you're a local chapter and you have students who come from economically weak backgrounds if they are doing NPTEL courses writing exams once they pass it we will refund 50% of the fee to them based on the college's recommendation so that is something that you can do also all this data that we give to the SPOC can be directly shared with bodies such as NAC and NBF and they come for accreditation and you can say that we are showing this data inside your dashboard that has also been very helpful for colleges is what we came to know these are local chapters that we have this is a state-wise distribution I think Maharashtra is the highest followed by Tam Nadu and first we had these telling Anna comes in maybe 7th position with 234 local chapters currently every semester we also we have some few local chapters outside of India and some countries they're not very active but yes they do keep using our courses as they can we also rate local chapters every semester based on the participation from their side in the particular semester so how many people have become topers how many people have written exams you know done well based on it we also rank and rate the college local chapters and we publish this list publicly on our website every semester other interactions with local chapter people we do soft skill training like I already spoke about it we take help of people here to go to exam centers as our representatives faculty can help in translation effect efforts and yes like I said we also rate it and we give recognition to the spark and the college also every semester so these are certificates that we give them based on the formula that we have to the college and the spark yeah these are other categories that we have on based on faculty participation based on how local chapters improve semester after semester there's a separate category part science colleges also yeah star LC who have been in the top 10 or top 100 over the last four years that's what we have done that's it as a presentation that we had today so I hope you learned about NPTEL about what we do here the NPTEL office the head office is in IIT Madras we have our head office here about 70 people work in NPTEL office in IIT Madras we have smaller offices in IIT Kanpur, IIT Kharagpur IIT Roorkee, IIT Gahati Bombay, Delhi and ISC Bangalore also and we all work together to keep this project going it's one of the most successful inter institutional project that eight institutes come together to work on this project and kind of run it without a break and to kind of offer 2000 courses that we have today that's what we want to say so my suggestion and you can say maybe I'm old enough to advise all the young students out here also is try out NPTEL courses today go and go to Swayam website drawing a few courses learn from it sample the videos there you may or may not be able to follow certain videos you may not like certain courses maybe some courses are difficult it's okay you can drop off it is not like a college course that once you join a course you have to be in that course you have to complete that course otherwise somebody will think something somebody will ask you something or maybe a paid fees that will get wasted there is nothing like that in an NPTEL course so you can join in in five courses you can join in in 10 courses try out week one videos you can see what you're interested in and then shortlist it to maybe one or two courses follow it up see where they are able to do assignments maybe it even comes down to one course in a semester but follow it till the end do all the assignments watch all the videos register for the exam do the exam and become a talker in it I think that's something that is very very doable I hope you were able to get that message as we presented in the last one and a half hours that is what we were trying to tell that don't be afraid that these courses are from IITs I cannot do it it will be very difficult there are 15 lakh people who have written exams with us so far it is not like all of them are coming out from IITs or they are all like very well accomplished they are all coming from highly studious background something like that they are also students just like you in any college what you need is the interest to study what you need is the motivation you have to put out time it is not like two days before the NPTEL exam let me study and get a certificate you will not get a certificate every week you have to watch videos every week you have to do the assignment then you can definitely get a certificate so you need hard work and it will pay off but it is not like I cannot do it there is no such thing as I cannot do an NPTEL course anybody can do an NPTEL course we showed you people from age 86 years doing NPTEL courses we showed you people from all nook and corners of the country doing NPTEL courses there is local language translation support of transcripts in some courses that will help you understand the subject better your faculty are there to help you if you do not understand something your friends are there to help you if you all join a course together discussion forum is there ask questions ask doubts if you have something there but try out and learn something which is outside of your college curriculum today when companies are looking for recruitment one of the things they see is okay your college degree is there apart from this what have you done that is something that all of them are looking at are you able to learn on your own are you able to study outside of college curriculum are you picking up any skills that are relevant for any job you might be trying for this is something that companies do look out for it's not just a resume saying BSE with this VA with this that might be one thing but then we look at okay what else what else what else that is something that all of us look out for when recruiting people so it might be good to have a few certifications to at least get your skills enhanced to learn what you are interested in to try and explore new opportunities that are there it is free there is no fee to learn you can download and watch the videos you can download and do the courses you can enroll in swine courses and do it only the final exam has a thousand rupees fee that also we are giving fee waivers if you're a local chapter so you can use that also after pay first get your exam done pass the course we will be able to help you with fee waivers on that front also so don't let money be a barrier don't let anything else be a barrier put in your time put in your effort learn from NPTEL get skilled and I hope you do much much better till then all the best and I hope that all of you do enroll to a course thank you hello sir yes ma'am you can try your activities okay ma'am is it visible to you yes sir it is visible already ma'am has given a very nice kind of presentation on the importance of an NPTEL in our curriculum okay so the benefits that particularly the gate after doing an NPTEL certification so you can perform a domain certification you can we have an FDP benefits you can we have job oriented courses soft skill courses and NPTEL stars okay so this is from our institute site okay so this is an onboard SPOC showing an active local chapter this is for our college okay so the total number of an NPTEL course certification done by an our institute from an academic year 2017-18 to academic year 2022-23 up to the first semester so the total number of an course certification done is in a 694 then this will be an I showing that when the category wise means the total number of a course category of an gold, silver, successful and also showing that the number of persons who are going to have a topper in that particularly the every time means our college and having the toppers also many of the category of gold and the elite and also the almost 95% times our active local chapter apart from this one okay our college has given a refund policy to faculty persons those who are scoring a 90% and above so we are refunding 100% exam fees and those who are in a category of 75% to 89% we are refunding a 75% amount of money and those who are having that in the score open a 64% we are the 50% of an exam fees to them also means the our college is having that one the swim NPTL local chapter running and actively we are having that one the smart TV and smart room for that to watching that one the swim NPTL videos and the courses okay so that's all from my side okay thank you thank you sir my screen is visible visible sir okay a very good afternoon I am Dr. Omkar Vaidya single point of contact person for swim NPTL local chapter of Sunderp Institute of Technology and Research Centre I welcome you all for this NPTL E awareness workshop for NPTL local chapter is established in 2017 with ID 1141 so we have this two URL one is given by NPTL and one is our institute own URL which can be seen here and all details regarding registration list of student who have passed the policies P-Weber schemes and achievements can be seen to everyone through this link we have started this local chapter with four objectives first is to make learning material available to students of engineering institution through easier means second is to provide students and faculty of platform to listen to the lectures delivered by IITs and IISC's update them and get certified for participating in online courses and assessment to make students employable in industry or pursue suitable higher education through swim NPTL online certification courses and last to own teaching of faculty members and use online courses material to supplement their teaching because we know that NPTL online courses has played a vital role in teaching learning process and especially in accreditation process of MBA and NAC and different framing frameworks like NIRF and area ranking right from inception we have been active local chapter for July December 2017, January June 2018, July December 2018 and recently July December 2020 to semester examination various initiatives of local chapter and very first is the awareness workshop how to motivate the student how to enroll maximum number of student and how to participate the maximum number of students in online courses as well as how to verify maximum number of exam registration so for that we have conducted in each year we have conducted this awareness workshop but this two workshop that has been highlighted which is basically in association with institution of this is with Ministry of Education Government of India that is IDC Innovation and Entrepreneurship Development Cell and Technoclub in 2019 and another is we have conducted in the induction program EICTE has given the guideline that first year engineering students we have to conduct the induction program we have given the awareness so maximum number of student has got awareness and they have enrolled and they have registered for examination then we have given NPTEL awareness workshop outside the institute that is in Sandip University in our campus school of science in 2022 and ensure the maximum participation from their institute also there we used to give felicity felicitations after each semester so these are the sample felicitations snapshots then we do the mentoring session during each semester examination so mentor is the important part of this NPTEL online courses so we assign different faculties from different subjects a mentor and mentor take a follow up of the students so these are the sample mentorship awareness session that has been conducted through this local chapter the unique initiative of this local chapter which is done only in this institute is the mapping of each subject of undergraduate program with NPTEL current and older semester so current online courses and older online courses which is available in archives of NPTEL we have mapped each and every subject for example 2nd year ENTC and 3rd year ENTC 2019 course each courses has been mapped with different available NPTEL online courses currently running rerunning or which is available older that is in Archive so students from different aspects they get the content from IIT's IISC faculty and they will not missed any part of the syllabus and they get a unique and very standard content from this course so we have disseminated this PDF file of the mapping of each subject of 2nd year, 3rd year and final year with the different engineering colleges those who are running this particular courses so these are the snapshots that is available with us another unique initiative is we provide 100% free waiver to faculty based on their outstanding performance in NPTEL examination so this is very very important then these are the course registration and examination statistics of our NPTEL local chapter right from inception so from July December 2020 we are in active local chapter average percentage of result is more than 75% so this is comprehensive statistics of registration and examination statistic these are the number of candidates who have enrolled the courses so from them we come to know how many registration more than thousands of registration of the courses has been done by our students apart up till so these are the achievements of our NPTEL local chapter who has been recognized in various local newspaper so we give appreciation to the students we give appreciation to faculty and this has been also observed by our management so thankful to the management thank you madam this is from my side and big thank you to the NPTEL thank you sir so we have disabled the YouTube comments both so we have posted the feedback in the description box so we don't have any Q&A session so we can go with what up thanks Omkar sir so why attract the issue with the amount Omkar sir you can give what up thanks sir very good afternoon to one and all present in this meet especially Dr. Bharti madam from IIT Madras Priyadarshani madam, Kamla madam, Divya madam the NPTEL IIT team from IIT Madras respected principal SITRC Dr. millionth party sir respected principal A.V. C.Y Sangam Dr. M.A. Venkates sir Bharat Borkar sir then Dr. Sonkar sir A.V. C.Y Sangam I first of all very thankful to Dr. Bharti madam for giving the or explaining the comprehensive awareness of NPTEL awareness and the different initiatives that has been conducted by NPTEL in terms of domain certification I am very glad to inform you madam that our some faculties are on the verge of completion of the domain certification they have only 2 to 3 certification remaining so thank you very much madam for giving the opportunity that the consistently those faculties or those students who are engaging this NPTEL on the courses and they are successfully completed the courses you have given the opportunity for a different certification that is called domain certification. Second is madam has explained nicely the NPTEL starts so NPTEL starts is basically promoting students who performed well up till and it is also very much inspirational to the other students because NPTEL starts of having different variants we have seen the different names that has been given by the for this NPTEL starts so again I am thankful to the NPTEL team to start this idea. Third then is the internship we know that the Texas instrument and some other industry we have means NPTEL has been collaborated so those students who have completed the particular certification courses the top students will be get the internship in that prestigious industry and in future that internship will turn in the placement. Another important initiative of NPTEL is I want to quote here is the gate preparation. NPTEL has provided a free of cost gate preparation and they have solved up till all gate question papers of different subjects and they have started the small small practice session on gate so gate preparation for that NPTEL has made a vital role so with this I am very much thankful to Bharati madam and everyone who is attending this e-awareness workshop I urge the faculty members and the students of different institute and please register for this NPTEL courses then attend the exam get certified get NPTEL stars then get a domain certification which will be helpful for their future endeavor. Thank you very much madam, over to you madam. Thank you sir, thank you very much and I wish you thank you for all and then thanks for joining this session you all me early in the meetings student employment workshop sir can you hear me no, sir, hello madam thank you for talking to sater sir yes sir, you may leave the meetings thank you for in giving an opportunity. Thank you, madam. Thank you, madam. Maybe you might, maybe we might interest. Sure, so what, yeah, what we do is, I, I'm the, I, I, I, I'm the, I'm the, I'm the, Hello madam, Abhijita, Madam.


good morning. In this lecture, we will be studying singular value decomposition. This topic embodies a very deep connection between quite a few different topic in the area of linear algebra. Consider this situation, we have already studied Eigenvalue problem in which we wanted to decompose a matrix A in this form with u and v equal. We have already studied Eigenvalue problem and all the time of our study in Eigenvalue problem, we have faced this question whether the decomposition of this sort will exist or not, if it exists then how to handle it and so on. So, it would be nice always in the Eigenvalue problem if we could make this lambda diagonal with u and v orthogonal and all such things. And at every step, our work was mired with the difficulties of several sorts. First among all matrices, we could ask this question only for those matrices which are square that is a subset. A subset of matrices, the subset of square matrices constitute the only matrices for which this question arises. So, for square matrices this question arises to begin with. Then in the square matrices not all matrices can be diagonalized, not all square matrices can be diagonalized. So, among square matrices we had a subset which is the set of diagonalizable matrices for which this kind of a decomposition is possible. Even among diagonalizable matrices, we had another subset of matrices which are symmetric for which this decomposition could be affected with orthogonal v which is same as u for that matter with that condition fulfilled. Even among the symmetric matrices for which we had this valuable theorem that you can work out an orthogonal diagonalization, even there the diagonal elements of lambda could be negative. Now, even among symmetric matrices we had a sub case subset which is positive semi definite in which case the lambda i turns out to be non-negative. Now, this is the best possible situation which we could think of and that is a sub case of the sub case of the sub case of the generalized general form of the matrices. Now, we can ask this question that we do not ask for a similarity transformation and we focus on this form of the decomposition. When we say we do not ask for similarity we basically want to allow u and v to be different. So, in that case we ask this question that if we do not ask for u and v to be equal, then what of these we can ask for and get results and with just this one relaxation of allowing u and v to be different, different in content as well as in size. If we allow that, then we can get a decomposition of this sort which is guaranteed for all matrices irrespective of size and shape that means even rectangular matrices with orthogonal u and v matrices and with non-negative diagonal entries in delta in this matrix lambda, diagonal matrix lambda. That is in that case we do not refer to it as lambda because lambda has been already used for the matrix of Eigen values. So, we show that as sigma that means that just by allowing this u and v to be different we can effect a decomposition of this sort with all the other facets that is the decomposition will be possible for all matrices and it will be always possible. The question will arise for all matrices including rectangular that diagonalizing that decomposition we cannot call it diagonalization that decomposition will be always possible with orthogonal u and v not same anymore and the diagonal entries of this matrix sigma will be all non-negative such a decomposition is the singular value decomposition and those diagonal entries are called singular values of matrix A. Underline is this very important theorem called the SVD theorem or singular value decomposition theorem. The theorem says for any real matrix A of size m by n there exist orthogonal matrices u which is m by m and v which is n by n both horizontal such that u transpose A v is diagonal matrix of size m by n. Now, what is this idea of a diagonal matrix of a rectangular size. So, its diagonal entries are sigma 1, sigma 2, sigma 3 etcetera all non-negative which you obtain by getting a square matrix first of size p by p in which p is lesser of the two dimensions m and n. Now, if you want this diagonal matrix to be m by n size then whichever is larger m or n that many extra rows 0 rows or that many extra 0 columns you append and these diagonal entries sigma 1 to sigma p are called the singular values of this matrix A. A similar result is there for complex matrices then for that the SVD theorem will read for any complex matrix A belonging to C m by n there exist unitary matrices u and v such that u star A v where star is a conjugate transpose is real sigma this is always real and so on. So, now this theorem gives the basis for the decomposition in this manner for a matrix A. Now, the question arises how to construct u v and sigma the three components the three factors the way we work out the construction at the same time provides the proof also of the SVD theorem that such factors u v sigma will always exist.  As we multiply it u being orthogonal u transpose u will be identity and we have got v sigma transpose sigma v transpose. Now, sigma transpose sigma we have already discussed that sigma is a matrix of this shape in which if m is less then it will have only m columns which will mean that the matrix will be only this much these rows will not be there. If m is less then it will have this shape if n is less then it will have this shape. So, extra 0 rows or extra 0 columns there will be no question of anything here because one of these 0 blocks will be there not both. So, if sigma is of this shape then sigma transpose sigma will be a square matrix in which the diagonal entries will be sigma 1 square sigma 2 square up to sigma p square and then since this matrix is n by n size. So, if n is larger then there will be additional 0 entries in the rest of the diagonal positions and all the of diagonal entries will be 0. So, that is the description of this sigma transpose sigma. Now, here this sigma transpose sigma this matrix is being called lambda which has a region. You see A transpose A is certainly symmetric not only symmetric it is positive semi definite also. You cannot say a priority whether it is positive definite or not, but positive semi definite it will certainly be. Now, if this is a symmetric matrix then this certainly has a diagonalization and orthogonal diagonalization for this matrix and this v lambda v transpose is actually the decomposition that you do when you solve the diagonalization problem of a symmetric matrix. So, that means this v which you want in single value composition is in fact the matrix of Eigen vectors of A transpose A and this lambda then is the diagonal matrix of Eigen values of A transpose A. If so then we already know how to determine v and lambda because we have studied the Eigen value problem of a symmetric matrix in good detail we can effect this diagonalization. So, that means by effecting the diagonalization of a symmetric matrix we determine v and lambda. The moment v and lambda are determined we can work out sigma because each diagonal entry of lambda the first p entries are nothing but sigma 1 square sigma 2 square sigma 3 square and up to sigma p square. So, from the first p lambdas from here which are all non negative we can take the square root. So, when you take the square root there are 2 square roots for a positive number 1 positive 1 negative. So, you collect only the positive ones which you put as sigma 1 sigma 2 sigma 3 etcetera up to sigma p. So, all the non trivial entries of this matrix sigma is in our hand then and then we append that with appropriate number of 0 rows or 0 columns depending upon what is the size of A which is the same as the size of sigma. So, that means v and sigma are now in our hand.  So, that means we can post multiply that original definition of the singular value composition with v and then v transpose v will become identity from here you will get only u sigma and on this side you will get A v in which in this entire equation A was originally given v and sigma we have determined and we are left with the problem of determining this matrix u the columns of the matrix u. So, 4 situations will arise when we go to determine the columns of matrix u. In fact, 4 situations may arise in any particular case only 3 of them will arise the either the third will arise or the fourth will arise depending upon whether the matrix A has more rows or more columns. So, first situation is actually the one in which you will have some information to determine.  That means you will get this kind of column equations when you break this column by column that will be the first column r columns if r is the rank that means for the non-zero singular values. So, out of these p singular values some of them may be 0. So, for non-zero singular values corresponding column equations will give you this kind of equations and if sigma k is non-zero then determining the corresponding columns of u is easy you just divide A v k by sigma and you get the columns of u. So, these columns developed from here are bound to be mutually orthogonal you can verify that suppose 2 columns u i and u j have been developed like this and you want to find out u i transpose u j. So, they are not only orthogonal they are also normal that is each of them is a unit vector also. So, for being also normal this has to be 1 if i and j is same and 0 if i and j is i and j are different. So, you can see this that when you consider u i transpose u j from these expressions from here you have determined u i u j then here you will find that you will get v i transpose A transpose A a v j now A transpose A is the matrix for which we actually solve the eigenvalue problem. So, v j is its eigenvector corresponding to eigenvalue lambda j that is sigma j square. So, when you write this here 1 by sigma i is here 1 by sigma j is here.   So, lambda j that is sigma j square is scalar which we can bring here and we are then left with v i transpose v j here. From there you find that if i and j are different then v i transpose v j is 0 because v and lambda together give the orthogonal diagonalization of A that means columns of v are mutually orthogonal. So, if i and j are different then v i transpose v j is 0 and you have got the orthogonality of u i transpose u j. On the other hand if i and j are same then v i transpose v j you will get which is 1 because v is orthogonal. So, each column v j in particular is of size 1. So, in that case v j transpose v j will be 1 and this sigma j square cancels with this sigma j square i is equal to j in this case. So, you will get 1 here that means u j transpose u j will be 1. So, that shows the orthogonality of all the columns that we have determined from this. This much for those singular values which are not 0 for non zero singular values. For the singular values which are 0 you have got this A v k equal to sigma k u k and sigma k is 0. That means you are talking about A v k equal to 0. So, the corresponding u k is left indeterminate. So, that means that you cannot determine u k from this relationship because the coefficient is 0, but it is left indeterminate. That means you are free to choose a suitable u k. What is a suitable u k? A unit vector that is orthogonal to all the other columns that we have already determined. Now, in a case where m is less than n that means u has less number of columns and v has more number of columns. That means in that case you will get further equations A v k for k greater than m for which on this side you will get 0. And from that there is no corresponding column of u to determine. So, this is gone. The fourth case is where m is greater that is the matrix A has more rows than columns. In that case after all these calculations there will be further row columns of u which are left indeterminate. So, just like the case 2 in this case also there are additional u columns, additional columns of u which are left indeterminate. So, just like this case in this case also the additional u vectors are determined to make the entire u matrix orthogonal. That means additional columns of this case with 0 singular values and additional u columns corresponding to this case with additional singular values additional columns which have no matching singular values. So, these 2 cases are determined based on the orthogonality requirement of u. So, that means in one line you can say we extend the columns of u determined from here to an orthonormal basis and that full set of m vectors will give you the square matrix u. So, this way after the 3 factors of the singular value composition have been constructed you have A equal to u sigma v transpose each of the c u have in hand. After constructing the singular value composition like this you would like to see what are the properties of such a decomposition. So, first question after verifying existence is uniqueness, is it unique? The actual answer is that it is actually not unique. For example, you can apply several changes in it and still the changed u sigma v will constitute another singular value composition of the same matrix. So, that means that you can do several changes. So, those changes are here and then you can say for a given matrix the SVT is unique up to these changes. That means it is actually not unique it is determinate, but such changes will not disturb the requirements. Such changes will not disturb the fact that the matrix is the decomposition is still an SVT of the given matrix. So, what are these changes which are possible? The same permutation of columns of u columns of v and diagonal elements of sigma. That means if you interchange sigma 2 and sigma 5 and at the same time interchange columns u 2 and u 5 and interchange v 2 and v 5 then the resulting u sigma and v will still give an SVT and so on. Now corresponding to equal singular values you have got columns of u and v. So, among them if you work out an orthogonal reorganization that is suppose sigma 2 and sigma 3 are same then you say that I will work out this. Now this will be my new u 2 and this will be my new u 3 and corresponding for v also between v 2 and v 3 also you will make the same transformation. This will be still the resulting u and v matrices with the same sigma will still give you a singular value composition which is valid. The particular case particular transformation that we have worked out here is cos theta sin theta sin theta cos theta here that is cos theta minus sin theta note that this is minus. So, that matrix is an orthogonal matrix. So, such orthogonal linear combinations for columns of u and corresponding columns of v is fine that will not disturb the singular value composition. For 0 or non-existent singular values you can do any linear combination any arbitrary orthonormal linear combinations among the columns of u or columns of v. So, that will still be all right. So, these reorganizations in an already existing SVD can be done and the result will be still an SVD. Now if this can be done then we can do something better than what we have done till now. That is we have determined sigma 1 sigma 2 sigma 3. Now if the permutations can be appropriated in that then we can order them that is we can organize columns of u and v in such a manner that the sigma value that comes first is the largest magnitude and so on this we can do. So, this is typically done when we work with singular value composition. So, that means the non-zero singular values come at the top with this order and after that the 0 singular values come and after that of course additional rows or columns may come depending upon the rectangular size and shape of the given matrix right. Now here what is R? R is the rank and this is a very simple result which you can immediately establish that is rank of the given matrix is the same as rank of sigma which is R here other properties. You would have already noticed that matrix of matrix A is of size m by n that means it maps vector from R n to R m right in which this is the domain and this is the co-domain right. Now you can see that v being an n by n R orthogonal matrix can give a basis which is orthonormal basis. The columns of v are actually n dimensional vectors and they are all mutually orthonormal. So, that means that the columns of v give us an orthonormal basis for the domain. Similarly, columns of u will give an orthonormal basis for the co-domain and now here we see how these new basis v and u decompose the domain and co-domain into orthogonal subspaces. So, you consider the application of A on an arbitrary vector x with A written as u sigma v transpose. Now if you represent the vector in the domain the vector x in the domain in this new basis v then the expression the coordinates of that those vectors in this new basis will be v transpose x actually v inverse x, but since v is orthogonal. So, it will be same as v transpose x right. So, if we call that y then we will have u sigma y u is written here and recognizing that sigma is a diagonal matrix with sigma 1, sigma 2, sigma p written on the diagonal entries among which the top r r non zero you will have sigma y as sigma 1, y 1, sigma 2, y 2 etcetera up to sigma r y r below that everything else is 0 right and u has been broken and written in this fashion r columns here and the rest of them here. Now when you consider this product you will find the product is sigma 1, y 1 into u 1 plus sigma 2, y 2 into u 2 and so on up to this after that everything else is 0. Now see what is happening in this sum you will notice that this has non zero components along only the first r columns in this product the components along u r plus 1, u r plus 2, u r plus 3 etcetera are all 0. That means that A x has non zero components along only the first r columns of u right. That means u has given us an orthonormal basis for the core domain in which the range the vectors A x are contained only with the first r columns of u. That means u gives an orthonormal basis for the core domain such that the range is exactly described by the first r members of u and the rest of them describe an orthogonal component of range orthogonal complement of range. So that means the entire core domain has been decomposed into two orthogonal sub spaces. The first one is the range which is described with the first r columns of u which are corresponding to the non zero singular values and the rest of them are components in the orthogonal complement of range which are not in the range. Similarly on the domain side if you see this v transpose x is y.  So the entries the coordinates in y y 1 y 2 y 3 are actually v 1 transpose x v 2 transpose x etcetera that is v k transpose x is y k that is the coordinate y k is found like this right. So that is component of x along the unit vector v k. So the full x is component of it along v 1 into the v 1 unit vector plus its component along v 2 into unit vector v 2 and so on like this. Now in this you will find that those vectors which are here only make a contribution in the A x mapping. Those here will not make any such contribution because y r plus 1 y r plus 2 etcetera are 0 that we have already seen right. They are made 0 by the in this product sigma y.    So you find that on the co-domain side range is constituted by the columns of u corresponding to non-zero singular values and on the domain side the null space is spanned by the other columns other columns of v that is columns of v which are corresponding to the 0 singular values or non-existent singular values and that is it. Now with this understanding in the background we proceed and find a few more interesting things. In particular we work out the revised definitions of norm of a matrix and the condition number of a matrix. In basis v if we write a vector in the domain in this manner then this can be written as v c where v is the matrix with columns v 1 v 2 v 3 etcetera up to v n and c is the vector with these scalar components. Then from the definition of norm which we have seen earlier in the chapter 7 of the textbook in an earlier lecture we discussed. So from the definition of the norm of a matrix we say that norm square is maximum over v of norm a v square by norm v square. Now in this if we insert this description of the general vector v that is v c. So then first of all from the norm definition we get this and there in place of small v we insert v c then we get this. For v we have v c and for v transpose we have c transpose v transpose. Now here we have already seen that a transpose a diagonalization was carried out with the basis matrix v and the corresponding diagonal matrix sigma transpose sigma. So in place of this whole thing we can write sigma transpose sigma. Now here sigma transpose sigma is a diagonal matrix with entries sigma 1 k square sigma 2 square up to sigma p square and then perhaps additional 0. So this numerator breaks down to basically this and now you say that we want the maximum of it. When will it be maximum? If sigma 1 sigma 2 sigma 3 sigma 4 are not all of the same magnitude then this will be maximum when c k is a vector when c is a vector in which the only component is along the largest one which gets magnified by the largest amount. Then only you will get the maximum value of this and so you get the norm as norm square as the case where only that c k as a non-zero value for which sigma k is maximum that is sigma max. So when you put sigma max there then you get this. So norm is now found as the largest singular value of the matrix. So this is the new revised definition of the norm of a matrix. Now for a non-singular square matrix we worked out condition numbers. So here again we try to do that for A inverse we get this which is v sigma inverse u transpose which is this. Now you notice that by the same definition if we try to work out the norm of A inverse then it will be the largest singular value of A inverse and the smallest singular value from A will actually in its reciprocal will give the largest singular value for A inverse. So you find that the norm of A inverse is 1 by sigma min of the original matrix A. So the condition number is norm of A into norm of A inverse that is sigma max into 1 by sigma min. So you get this and that brings us to the revised definition for norm and condition number of a matrix. The new revised definition of norm and condition number will be like this. The norm of a matrix is the largest singular value and the condition number is the ratio of the largest singular value to the least. Now note that this revised definition of condition number can equally cater to rectangular matrices also. The old definition based on inverse would not be able to do that. Now note one more important issue. If you can arrange the singular values in decreasing order as we have been talking about then with rank of the matrix as R you can write it in this manner. So in which U R is that sub matrix which has all the columns of U which are corresponding to non-zero singular value.  In that case this matrix A which is U sigma V transpose can be multiplied in this block form in which the three components that you get out of it will be 0 because of these and the non-zero component is only this U sigma U R sigma R V R transpose. So the other components are 0 and this gives you this summation. That will mean that if you can store the components of U and V the columns of U and V which are corresponding to non-zero sigma then that alone along with the sigma values of non-zero sigma sigma k's will be able to reconstruct the matrix A. And that means that for a large matrix with only a few top singular values as non-zero and significant you can effect every efficient storage and reconstruction. So with this background now we go ahead and see what is the application and what is the particular advantage of singular value decomposition for solving linear system of equations A x equal to b. And we again revise the definition of pseudo inverse compared to what we did earlier in the chapter 7. So in the background there is this term called generalized inverse. For any matrix you can define a generalized inverse or G inverse if for a vector b in the range A G B is a solution of this. That is for a matrix A a matrix G can be considered a an inverse of some sort generalized inverse if for a consistent right side vector b G B gives you the solution that way G operates something like an inverse. So pseudo inverse is actually a special case of generalized inverse the pseudo inverse or the Moore Penrose inverse is defined in this manner. And in order to differentiate it from the ordinary inverse we write it with this symbol A hash. So A hash is U sigma V transpose hash. Now here wherever an inverse is actually possible we take the A hash A we take the pseudo inverse as the same as the actual inverse. So the pseudo inverse of this will be V transpose hash sigma hash U hash. Now V transpose and U are orthogonal so for them actual inverse exist. So for the V transpose hash we write V transpose inverse which is V and similarly for U hash which is U inverse which is U transpose. That actual problem with actual problem is with this right. So this is the one which requires a definition that is defined like this. For this structure of sigma in which there is a diagonal matrix of R by R size here with R non-zero single values and everything else is 0 sigma hash is defined as this. So now that will mean that those diagonal entries which are non-zero their reciprocals will come here and those diagonal entries which are 0 so their reciprocals rather than infinity will put 0 here. This is very interesting. In place of 1 by 0 which should come as 1 by 0 by the ordinary rule here we are actually writing 0. So this is how we define the pseudo inverse or Moore-Penrose inverse. In elaboration you can write sigma hash in this manner. So sigma 1 2 so in place of the diagonal entries rho 1 to rho p you write where rho k is the reciprocal of sigma k when sigma k is non-zero and sometimes in practical cases even if sigma k is very small then we consider it as good as 0 that is here. So for those cases where sigma k is 0 or extremely small we put rho k as 0. So rather than putting 1 by extremely small number or 1 by 0 we actually put it 0 there. So this is the definition of pseudo inverse. Now sometime at leisure you should compare this expression and this description of the pseudo inverse with the special cases full rank cases which we worked out in chapter 7 as right inverse and left inverse. So in those cases where the matrix is full rank those definitions will appear as special cases of this. Now what are the inverse like properties or qualities of this pseudo inverse? First is pseudo inverse of the pseudo inverse of the matrix is the original matrix considering only actual 0 cases being put 0 here and not the truncations. Second important point which is like inverse that if a is actually invertible if it is a square non-singular matrix then this will boil down to the ordinary inverse and a hash b will give the correct unique solution of a x equal to b. On the other hand if the situation is not so good and if a x equal to b is an under determined but consistent system that is full rank case of more unknowns and less equations then a hash b selects that solution x star which has the minimum norm out of an infinite possible solutions. On the other hand if the system is inconsistent then this a hash b defined with the same formula then this a hash b will minimize the least square error that is if the system is inconsistent there is bound to be some error a x equal to b in a x equal to b a x will never be exactly equal to b then this same a hash b will find you an x star which gives the minimum error. Now if that minimum error giving solution is also not unique if there are infinite of them then at the same time it will give you that solution out of those infinite possible solutions giving the minimum error which has the least size. So all these sensible things the pseudo inverse does with the help of a single definition. Now you should contrast this with the solution which we obtained earlier from Kikhanov regularization. So pseudo inverse solution is typically used when you want precise values and also for diagnosing a linear system whether it has any such inconsistency or under deterministic problems and so on. On the other hand Kikhanov solution can be used when the coefficient matrix a changes over a domain and you want continuity of solution. So Kikhanov solution is preferable for continuity but for diagnosis and for precise solutions pseudo inverse solution is better. Kikhanov solution will always inhibit some error. Now in the exercises of this chapter in the textbook actually there is an exercise which asks you to determine the Kikhanov solution and the pseudo inverse solution and compare them for a matrix a which has one of the components variable. Now we want to know how this whole thing is accomplished by a single formula. So for that first we note down what is the pseudo inverse solution that we find that is this is the pseudo inverse of a and when we multiply it with b we get this sum where the summation is over K from 1 to r that is for all the non-zero singular values.  We can if we write it like this then we will find that the pseudo inverse solution that we are getting is actually a linear combination of r basis members V 1 to V r the corresponding components are these scalar values written in the parenthesis. Now we want to pose the problem as first minimization of the error and then if the solution is infinite then further minimization of the size of the solution and then see whether we get this same solution. So if we want to minimize the error this square error half norm square of the error a x minus b then as we open this we have already encountered this earlier once then the minimality condition first order condition is that its derivative its gradient with respect to x must be 0. So when we do that we get this as we got last time here now in place of a we write U sigma V transpose and through a few steps we come to this point. Now note that this is a matrix vector equation and this is the corresponding scalar equation for each component of that vector equation. So this is for each k from k equal to 1 to r where r is the rank that is non zero singular value. So from here you find that V k transpose x that is component of x along the unit vector V k turns out to be U k transpose b divided by sigma k. This sigma k square goes down in the denominator and this is what is actually sitting here. So in this solution x star is composed of several vectors V 1 V 2 V 3 up to V r in which the component of V k is this. So that means x star is actually giving you this combination of these vectors with these components. Now this first order condition for the minimality of this tells you what should be the components of the solution along the basis vectors V 1 V 2 V 3 up to V r along V r plus 1 V r plus 2 V r plus 3 what should be the component that is not mentioned here. That means those components can be anything the error is still minimum because the condition is satisfied. That means the general solution for minimum error you can constitute with the components along V 1 to V r as specified here and any component along the rest of the directions that will give you this with components as prescribed along the first r directions along the first r basis members and anything in the rest that means y is free here.  So these y 1 y 2 y 3 can be anything the V bar is the basis for the null space. That you will appreciate because any null space member will not change anything in the solution in the right hand side. So now we say that out of all these infinite possible solutions which one is the one which is of least size. So then what we ask for? We ask for how to minimize the size of the vector subject to this error being minimum anyway. That means the solution you take from here and minimize it with respect to y that is which y to select to minimize the size of the vector this. So we say minimize the size that is x norm square that is this. Now you find that x star this part is a linear combination of V 1 to V r and this part is a linear combination of other basis members and all other basis members are orthogonal to the basis members of the first family. That will make this x star sitting in one subspace and this part V bar y sitting in another subspace two subspaces being orthogonal to each other. So how do you find the square of some of this? If the two members are in a in orthogonal subspaces. So since they are mutually orthogonal this will be simply x star square plus V bar y norm square. Now you find that if we then want to ask that which y will give this as minimum where this is already available and cannot be tampered only y can be changed then y equal to 0 will give you this as 0 and this sum as minimum. So that means that y equal to 0 will give you the minimum size vector x which is of this form which minimizes the error. Now how this whole thing happens that you get all the optimal conditions in the solution that you construct with the help of the pseudo inverse. So for that let us investigate the anatomy of this optimization through SVD. If we use basis V and basis U for the domain and co-domain then the variables x and b under question x unknown b right hand side known. They are transformed as this that is in the new basis V the expression of x will be this y and in the new basis U for the co-domain the vector b will be represented as t which is U transpose b.  So then you basically get the equation in the new basis V on this side and U on this side as sigma y equal to c. This is a completely decoupled system because if we write this system of equations sigma y is equal to c we will find sigma 1 sigma 2 up to sigma r like this y 1 y 2 up to y r and below that possibly more variables up to y n and on this side we will have c 1 c 2 up to c r and below perhaps more things. Now the way the solution has been constructed you get the useful information only from the first r rows first r equations and they are completely decoupled because y 1 is simply c 1 by sigma 1 y 2 is simply c 2 by sigma 2 up to this y r is c r by sigma r. What happens below below you find that all 0's here that means right left side of the equation will give 0 question is what is here in c. If there are corresponding 0's here then that will mean that the system is consistent but that information 0 equal to 0 is completely unusable it does not have any information content. On the other hand if some particular values here are non 0 that will mean we are talking about 0 equal to something non 0 that means that is the conflict that is the source of inconsistency in the system of equations. So in this situation we find that for k equal to 1 to r this is what we determine and that is the only usable component. And for c k greater than 0 for k larger than 1 r that is below for c k non 0 for c k non 0 you will find that you have purely unresolvable conflict. So that is simply the inconsistency decomposed into a an orthogonal subspace and which cannot be compensated for by any other component. And c k equal to 0 will give you completely redundant information that is again the completely redundant information is also collected over an orthogonal subspace which cannot be changed from any other component from outside. So by setting the appropriate diagonal entries of sigma hash as 0 SVD extracts this pure redundancy and inconsistency and rejects that. So it rejects the redundancy it rejects the inconsistency and gives you that solution which is the best possible achievable. At the same time since these were free still because the usable component gave you values upon this much setting this all these variables as 0 minimizes the norm of y and since the norm of x will be the norm of v y v is orthogonal. So through the multiplication of an orthogonal matrix the norm of the factor does not change. So minimum y will mean minimum x for the norm. Now the points to notice here important points to note are the following. First SVD provides you a complete orthogonal decomposition of the domain and co domain and it separates functionally distinct subspaces. On this side the null space from the rest on that side the range from the rest. It offers a complete diagnosis of the pathologies of a system of linear equations. And then zero inverse solution A hash B gives you the most important solution of a linear system in all cases. Apart from these what has not been noticed till now very clearly is that with the existence of SVD guaranteed that any matrix real or complex you can write as u sigma v star or u sigma v transpose. Many important mathematical results and many other formulations can be worked out in a very straight forward and direct manner. In many of the cases in coming lectures based on this existence of SVD only you will find that you will be able to appreciate the deductions of many of the result quite easily. So here we in this lecture we have actually connected two important problems systems of linear equations and iron will problems together through the singular value decomposition. In the next lecture which will be the last lecture of our linear algebra module we consolidate a few important issues based on the abstract fundamental ideas of linear transformation. Thank you.


A British Prime Minister Boris Johnson has apologised and vowed to fix it, that these were after a report on Monday into lockdown parties held at his Downing Street residence, a criticised Sarah's failures at the heart of government, a report by senior civil servant Silgway into lockdown, breaking gatherings at Downing Street under Johnson condemned some of the behaviour in government has been difficult to justify. The Justin said the government hard to learn from the criticism raised and that he would make changes to his Downing Street operation. I understand the anger that people feel, but Mr Speaker it isn't enough to say sorry. This is a moment when we must look at ourselves in the mirror and we must learn and while the Metropolitan Police must yet complete their investigation and that means there are no details of specific events in Silgway's report, I of course accept Silgway's general findings in full and above all her recommendation that we must learn from these events an act now. Mr Speaker, I get it and I will fix it and I want to say to the people of this country, I know what the issue is, it's whether this government can be trusted to deliver and I say Mr Speaker yes we can be trusted, yes we can be trusted to deliver.