Okay, let's go ahead and get started here in the interest of time. Hi, how's everybody doing? Good, good, good camps so far today. Thank you for coming to my session. This session is called Making Security Make Sense to Users and Clients. So first I'd like to get to know everybody in the room just a little bit. How many of you are actively building websites for clients? Ooh, nice. And how many of you are managing those sites on a monthly basis with a maintenance plan or retainer or some sort? Great. So my intention with this talk is to help provide a few useful tips for growing your business and also adding extra value to your client as it pertains to website security. So I am Adam Warner. I am the open source community manager for SiteLock. It just so happens it's a website security company. I am also the owner of Foo Plugins Plugin Business. I'm passionate about website security and you're going to learn the exact reason why. I'm passionate about security pretty soon and I'm all of these other things. So today I want to cover securing your own site first and why that's important. Securing your client sites, the benefits of your business for doing that, communicating security and the benefits efficiently to your clients including security in the project scope, security best practices that you can implement for yourself or for your clients today and security in your maintenance programs and pricing and reporting options. So let's first talk about why securing your own site is a good first step. How many of us have actively secured our WordPress sites? Great. How many of us are using plugins to do that? How many of us have done that manually through some manipulation of HT access or other things? Great. Okay. So the first reason to secure your own site is really about your reputation as a web development provider. So website hacking attempts happen all day every day and I'll talk more about the how and why of that in a minute. So when I talk about the reputation of your business, imagine that someone is looking for a web developer in Hamilton and they Google web developer Hamilton and they get to your site and Google throws a warning that this site may be hacked. Not very good for your reputation, right? Yep. Oh, thank you. So what are you going to do when that happens? You're going to leave that site immediately. You're not going to trust that person or that business with your own business. The second reason is familiarity with security best practices. So I'd never recommend a service or something to my clients that I haven't used myself with one correction. I mean, I would never do that again. I've been burned by that. I failed clients in the past because I didn't do proper due diligence on the plugin or the theme or the service that I was recommending to them. And so I've since learned my lesson. So the point of becoming familiar with security best practices is really about eating your own dog food, right? If you're going to make suggestions to your clients about how to secure their sites, you should be familiar with A, how to do that. And B, you should be securing your own stuff. And it protects your business. And that's kind of an obvious one, right? It protects your site from getting hacked. Or if you do get hacked, you have a better plan for getting your site back up more quickly. So we have a lot of goals as web development providers, but first and foremost should be protecting our own business. And I'm going to tell you a true personal story of what happened to me and why I became passionate about website security. In 2006 and 2007, I started a WordPress multi-site installation called Indelab. And it was a site geared toward providing websites for creatives, for musicians, for artists, painters, what have you. And it was all free. You could sign up. If you're not familiar with multi-site, it's basically what runs WordPress.com. When you go to a WordPress installation, you can sign up for your own site underneath that. So it's adam.domainname.com. So it was all free.  And then I started to monetize it by adding extra features if you were paying a monthly fee. At the time, I was working full-time for an audiobook publisher. And this business started to take off. People started to pay. You liked the features that we were building in. And I was really, really close to quitting my full-time job. And then one morning, a bunch of emails came in. I can't get to my site. My site is hacked. It has all this garbage on it. It's redirecting to Viagra and Cialis ads. So I dug in. I took that day off of work, actually, dug into that. And at the time, there were no security plugins. There were no security services where I could just call and have them clean it up. So I was on the WordPress and WordPress multi-site forums trying to get help and then emailing individual people. And I had it cleaned. And I thought I was good to go. And the next day, everything was reinfected again. So I went through this for about a week, back and forth. Meanwhile, answering customers saying, hey, this is what's going on, communicating to them. Well, it went on for more than a week. And we were approaching about three weeks of this. And I was frustrated. I was a bit depressed because why can't I fix this? And of course, as you can imagine, my customers were not very happy, right? So what I ended up doing is closing the business, refunding everybody, and calling it a learning lesson.   And what can I provide people, right? But that led to my interest in website security, which is now my full-time job. So I hope that doesn't happen to you. And I hope that that shows you the importance of website security, not in terms of just having a website that doesn't look bad or that's redirecting somewhere, but protecting your business and the business of your clients. So let's talk a little bit about securing your client's sites. Your site is secured. We're good to go. And are you actively implementing the basic security best practices on the sites before you hand them over? Is anybody doing that now and doing basic security steps before you hand over to a client? Great. Good stuff. So the first reason to secure your client's sites is that it's in your best interest. Has anyone ever received an email or a phone call on a Saturday night at midnight that says, urgent, my site is hacked, help me or some other thing, right? Many of us have. I have heard those exact words, as I just told you. So managing your client communications can be taxing when those emergencies happen, right?   You know that many of us, if not all of us, basically work 24-7 when we're building our own businesses. But if you have these mitigation steps in place, you will have more time for yourself and therefore it is in your best interest. It also is peace of mind for you and for your clients. So whether it's protecting sites proactively or giving them the education and tools that they need to do it themselves, the end result is you can relax a little bit and so can they. So let's talk about education for a minute. Has anybody tried to communicate to their clients the importance of website security? I have and is it safe to say that the typical response might be, I just blog about my cats, I don't need website security or it's too expensive and why would anybody want to hack my site or that's way too technical and it's beyond me and my eyes are glazing over and I'm not going to even listen anymore, right? So educating your clients or potential clients about website security to me isn't just the right thing for your business, it's the right thing to do period for the betterment of the internet, right? It can also set yourself, set you apart from the rest of the web development crowd. And what I'm getting at is security awareness in general but security awareness very specifically and I'm going to say this again a little later on but when you are first contacted by a potential client whether it's email, phone, my advice to you and my hope is that you will start discussing website security from the very start and not to overwhelm them but just to mention that as the first thing, one of the first things you talk about because by doing that you're going to make someone's security wear, whether they choose to go with you or not but you're also going to set up yourself and your client's site to talk about security through that entire process from the initial contact to the project scope to the proposal stage and the benefit there is obviously for your client's sites for you but there's a potential for more revenue for your business as well. So I look at it as we're the ones who understand how the internet works and therefore it's our responsibility to help secure that. So who is responsible for security? Can anybody take a guess? There you go. Ding, ding, ding. The short answer is all three. Ourselves, your clients, the web host, but to varying degrees. So I'd like to use an apartment analogy. So as providers, as developers, as freelancers, agencies, we're the ones that are creating the apartment building, right? We're the construction company. It's our responsibility to make sure that that website structure is solid. It meets all the building codes. There's no exposed wires. In other words, making sure that building, that website doesn't fall down. Now, web hosts, on the other hand, are primarily responsible for the security and maintenance of their servers, which is like an apartment complex where that building stands, right? They want to make sure that the buildings are protected and safe, all the sites within their network. They want to make sure that the access gates and codes are in working order. They want to make sure that the parking lots are lit and safe, the snow is shoveled, et cetera. As a website owner, the security and maintenance of your website is really your responsibility, just like an individual apartment would be within that building, within that complex. So in terms of communicating this to your clients, you can use that analogy, but also just like if they had a brick and mortar store, excuse me, I'll almost fill over, just like if they had a brick and mortar store, they wouldn't leave for the night and leave the doors open and the windows unlocked, just like you wouldn't do that with an apartment, right? So it's your responsibility, or your client's responsibility, excuse me, to make sure that everything is safe and as locked down as it can be. So I talked a little bit about setting your business apart and mentioning security, but also in educating your potential clients about security. So again, you can start from the very initial email or phone call and then expand that into education of what website security is as it pertains to their business. Again, the brick and mortar store, doors and windows locked. And then security, education and awareness really starts at the very beginning and should be a theme throughout the entire project and then into the monthly maintenance stage. And so even if they don't move forward again with your proposal, at least you've done your part to spread that security awareness. I mentioned additional revenue. There's additional revenue in the form of higher prices and residual income. If you are separating yourself as a business that is focused on the security of the internet as a whole, on the security of your client's business, you immediately separate yourself from the rest of the pack. And I wouldn't recommend but I would offer that you might even make security a required portion of your project when working with you because it shows your clients that you are serious, that you understand the internet and that you're a partner in their business rather than just trying to upsell them on some extra feature that they may or may not need because everyone needs website security. And in doing that, you can demand higher prices for the initial project. So if maybe now you're doing minimum projects of $1,000, maybe some of you are doing $100,000, but if you're doing a minimum amount that is around $1,000 or whatever it is, you can immediately demand higher prices in order to help set yourself apart too. So the key is to provide immense value that they can't get anywhere else. Now, more revenue can also come in the form of residual income, and that again is your monthly maintenance program. You can build in security reporting, scanning, web application firewalls, all kinds of stuff in various plans. And you can even use affiliate commissions to any security company that exists out there. Most of them have affiliate programs where if you can't sell your client on that and they come to you six months later and say, my site is hacked, then you can go get cleaned up with any of the number of security companies that are out there and make some money on the back end. So the struggle is communicating the need for that security effectively, right, to the average business owner who may or may not be technically minded. And we've talked about the typical reactions. It's too technical. It's too expensive. I don't need it for my five page brochure site. So how do you explain the subject of website security into terms that your clients will easily digest and understand or want to understand? So for me, I always communicate these three things. If you break down website security into the most basic questions, it's much easier to understand and communicate, and it helps you communicate that importance and the concepts and the terms in the way that's familiar to them. So the first is why. Why websites get hacked? Does anybody have a quick answer or opinion on why websites get hacked? Like the most popular reason. Say again? That's why. Yep. But why do hackers do what they do? Yes. User data? I was going to say marketing or advertising for whatever. Yep. Yep. Yes. The answer is all of those things. Yes. So they could do it for a little bit of 15 minutes of internet fame. You've been hacked by, you know, so and so. That would be known as a defacement where it's very visible. So a quick side note. We had a presidential election in the U.S. I'm sure everybody is aware. And one of the candidates' websites was hacked, and there was this query string you could put in the top in the big banner that had their slogan on it could be changed. And it was changed to a lot of different things. And one of the nicer things was I like turtles in reference to an internet meme. So there's reasons why people do that, right? For fun, for a little bit of fame. But by far, the most popular reason that a website gets hacked is for financial gain. All right? So one thing to know about website hacks is the overwhelming majority of them are automated. They're automated scripts that people take, tweak, or write themselves, and then they put out into the wild. And what their scripts are looking for are open doors. And open doors can come in many forms. It can come in the form of outdated software. It could come in the form of outdated web server software. And the reason why these scripts exist is a lot of times people will find companies that will pay for just clicks. So if you get to a site and you see what's known as a redirect, you go to a website, you get redirected to a CLS website or excuse me, a pharmacy website. The person who has created that script is getting whatever, a penny, two pennies per click or less. But now imagine that there's hundreds, thousands, hundreds of thousands of websites where people are getting redirected. It's easy money for them, right? So explaining to your clients why website hacks happen and putting it in terms of revenue and why it's an incentive, then you blow a lot of those misconceptions out of the water. Why would someone want to hack my cat blog? Well, because it exists and because there's an open door and because there's a whole bunch of other websites on that same web server that once I get in, it's not easy but it's easier to cross-contaminate rather than keep finding sites that have holes in them. Anybody here of Equifax? Yeah, Equifax, the Equifax hack gave access to personal data, social security numbers, addresses, family members to 143 million Americans and people from other countries. Anybody know how that hack happened? Outdated software. Outdated software on the web server that was hosting the Equifax website. And to rub it in just a little more, that outdated software was known about four months before the hack happened. So I'll get back to that in a minute. So who and how? So when we think of hackers, the stereotype is that it's some angsty, antisocial kid in the basement of their mom's house that's super smart, right? And targeting individual websites. Susie was mean to me. I'm going to hack her website. Well, there probably are stereotypical hackers out there wearing dark hoodies and such. But the majority, again, of those website attacks are performed by those automated bots. So how do these happen? They happen in a lot of different ways. And it all comes down to vulnerabilities found at various access points. Access points can include outdated software. They can include passwords, weak passwords, or newly discovered vulnerabilities in up-to-date software. So again, communicating the how and why to your clients in a simpler way is going to help them understand, again, the importance of website security. So also communicating when website attacks happen. Unfortunately, it's not a question of when a website will experience an attack. Excuse me, not a question of when, because hacking attempts happen all day, every day. We recently published our quarterly security report, and we found that every website on the Internet had an average of 44 attacks per day. Again, from these automated bots, which, for my personal website, is an average of 16,000 attacks every year. This is an automated gift of, I have to go back, North Security. And this is just an example of a few seconds' worth of attacks that are happening every second of the day. So now that you've communicated the why's and the how's to your clients, let's talk about how to implement some simple website security best practices. One are backups. Is there anyone here who doesn't know what a backup is? And that's okay if you don't. Okay, so you're basically creating a copy of your site and your database and your files, and you're doing that on a weekly and or monthly basis to have a good copy of your site if the worst happens. Doing this ensures that, of course, if something goes wrong, you can fix it. Updates. We just talked about outdated software, and I'm talking about WordPress Core. I'm talking about plugins. I'm talking about themes. But one thing that some people miss, and I missed it myself, is any other software that's running on that same web server that your WordPress installation is on. I am a shiny object guy. I'm doing this a lot for different applications.  So it's really important to keep all your software up to date. One thing to remember is to always run a backup before you update your software. I don't know if anyone else has experienced this with WordPress specifically, I've seen the white screen of death more than once because of an update. Maybe it was a plugin conflict or something like that. So number one backups, number two updates. Number three, strong and unique passwords for every single login that you have. Starting with your local machine, maybe even starting with your home Wi-Fi router, all the way up to every single site, every account that you have on the internet, strong and unique passwords. And I know what you're thinking, how in the world am I ever going to do that if you just choose one really hard password and then reuse it everywhere? Will that be enough? No, it won't. You have to use strong and unique passwords. I put a URL there on the screen. If any of you have your laptops out, feel free to load that. Have I beenponed.com forward slash passwords. And I will show you the outcome of that. This is what happens when I put in a password that I used to use quite often. You can see there, uh-oh, this password's been used, or seen two times before. It has previously appeared in a data breach. I have a pretty good guess of where that came from. Yahoo got hacked several years ago, 400 million accounts or something like that. And I think LinkedIn also got hacked. And I used that same password in a few different places. So, what is the answer to creating strong and unique passwords for every single site and remembering all of those? The answer are password managers. One password is a tool that I would recommend. LastPass is also a tool that I recommend. It helps you create really strong passwords, helps you remember them. Now, Grain of Salt. Website security is about reducing your attack radius, right? So, you have to have a master password for a password manager. What happens if your password manager gets hacked? It's happened. But at least you're reducing that attack radius by using one tool to manage all of the other ones, right? Number four are firewalls and CDNs to some degree. If you aren't familiar with a firewall, there are two types of firewalls. There are network firewalls and there are web application firewalls. A network firewall is something that your host typically has to protect their own network from web server to web server. And what firewalls do is they recognize and block known bad traffic. A web application firewall is something that as a website owner or your clients, or you would help your clients do that, it's something that you would put on your own site. So, the way a firewall works or a web application firewall specifically, someone's in their browser here and they want to get to your website over here on your web server. A web application firewall is basically a hardware and software program that sits in the middle and again recognizes and automatically blocks that bad traffic. Any provider that provides a web application firewall is constantly adding new rules to that firewall when there's new malware out there and detected and 100% matchable. So, firewalls and CDNs, I would choose a web application firewall of some sort even if you are not continuously monitoring because you're going to lower your bandwidth and your hosting costs just by blocking a lot of traffic. Quick side note, I had a WordPress tutorial blog back in the day and I thought I was the bee's knees. I thought I was so popular because I had tens of thousands of visits every month and then I installed a web application firewall and my traffic went hee hee because a lot of that was from automated bots. So, something to be aware of. Continuous monitoring, what I'm talking about with continuous monitoring is some kind of malware scanner. There are a bunch out there and they all do good stuff and you can continuously monitor your site to look at it as a person would or as a search engine would. So, different types of hacks have different signatures but if you're continuously monitoring every day, then it's more likely that you will discover that there's an issue and be able to fix it before Google recognizes there's an issue and blacklists your site and then your client's revenue goes down. So, that's what we're trying to do at the end of the day is protect your client's business and show them that you have the know-how and skill to do that and that you're a partner in their business. Including security in the project scope. So, you're going to hear me repeat the same thing. Just like discussing security from the first contact including the importance and requirement possibly for security best practices within the project scope can benefit your reputation and the revenue growth of your business. And this can be as simple as adding some line items in that says we've done X, Y and Z. It could be we've installed this security plugin and it monitors this and that. It could be something as advanced as we've taken these steps in the WP config file and the HD access file and we've whitelisted or blacklisted a bunch of IPs based on what your business needs are. It could run the gamut but if you're calling out that those security steps within the proposal you have the second chance there for that visibility of your focus on security. It gives you a more professional image because again you're separating yourself from the rest of the pack who maybe haven't even mentioned security and you're building trust as a partner in their business. You're saying it's important to me that I build something that lasts a long time and meets your business goals. I'm not just going to build you a website handed over and then on to the next. I want to have a long-term relationship with you. The benefits of including security as a service. Security from the first contact again in the project scope sets you up to demand that higher price for the initial build but in addition it also sets you up to include security options as part of your maintenance program or as add-on services. Again if they choose not to do any security other than what you've already built in they come to you the worst happens then you can offer those add-on services for a price and use some other vendor to actually fix it. Maintenance plans. We already have the answer to how many are offering maintenance plan which kudos to everyone who's doing that. That's the way to stay sustainable with recurring revenue. It's the way to forecast your business growth. So you could easily again run this security under your maintenance plans under the umbrella of backups and updates even or call out specific levels of security. So you could have it scanned every 24 hours. You could have a scanner built in that scans continuously more for enterprise clients. Those add-on services and some more examples of that would be the one-time cleanup. It could be the web application firewall setup if they decide to go with a vendor for that you could be the one to set that up. It involves some DNS changes to your domain to route traffic through. Now, automating maintenance and reporting. So if you're automating, if you're offering your monthly maintenance, you're likely offering backups and software updates. So how many of us are doing that manually? Are going into individual client sites and updating, right? And checking, okay, just a few. And how many of us are using a maintenance and reporting tool? Okay, a few more. A couple of really popular and good ones, if you don't know, are ManageWP and Watchful or Watchfully is the address. So basically, if you're not familiar with these, you can connect multiple single websites to either of these accounts and then you can bulk update software on multiple websites. You can bulk update plugins, themes. Both of these offer reporting tools.  These are all the updates that were done this month. And luckily, WordPress Core updates quite often, plugins update quite often. So there's probably, it's probably very rare where you won't have something to report to your client. But include that with the reporting from any security service that you may happen to use or a security plugin will be included with the updates. It shows your client that you, again, are serious, it's visibility to that security on an ongoing basis. So hopefully, they'll refer you to their friends and colleagues. So the benefits of a summary of a presentation. Again, this is only my advice. Secure your own site first. Learn the why, how, who and when of website security. Communicate the business benefits of that effectively to your clients. Included in the project scope, offers an optional or required service and automate the maintenance and reporting of your monthly plans and your security outcome. I'm happy to take any questions if any of you have any. Yes, sir. So the other options would be to use a different, oh, repeat the question. Do I have any advice for other security options other than site lock or security because you have clients who've been burned by both? Okay. My best advice would be to use another security option found within the plugin repo. That would be in the form of a plugin, not a cloud-based service. I honestly don't know. There's us. There's security. There's iTheme security. There's WordFence. My best advice would be to try the others that you didn't have a bad experience with or they didn't have a bad experience with or call us and let us know or call security and let them know about that specific situation and see if... Yeah. Sure. That's not the first time I've heard that. So there are, we do partner, specifically we partner with a lot of different hosts and I would say that that was probably a communication issue more than a technical issue and I can tell you that that is known and that has been worked on. So in terms of that specifically, that would be my response to that but if you're looking for a different security vendor, I believe the ones that I had already listed are the largest ones in this space. If anybody knows of any other providers, call them out. No. There you go. Okay, any other questions? Yeah, and I'm happy to talk with you further about that specific situation this afternoon too. Sure. Yes, sir? Yeah, the question was, can I talk about the differences between services like Watchful and any other security... and a security vendor. So the main difference is that Watchful and ManageWP, those services exist to manage multiple single installations of WordPress. I think Watchful handles Jumla as well and it handles the updates and backups but it doesn't... and both of them include security scans, a light scan, which then again looks at your site as a browser would see it. So it's more for management of multiple areas of individual websites rather than a strict focus on security. Okay, yeah. So a combination of both of those services would be better than one or the other? Well, it all comes down to your personal preferences and the needs of the business but I would argue that, yes, that both working together would be complementary. Just like in terms of the company I work for, there's complimentary features and benefits to having a security plugin and a security service as well. So in terms of site management, yeah, I don't think there's any reason why you wouldn't want to use that with some other security option. Yes, sir? Are there any websites or bloggers that you recommend for information on how to get started on this before you're in console, et cetera? Sure, yeah, there's a few. There's WP White Security as a guy named, I can't remember his first name, Robert Abela. He's out of Malta and he is an avid security person, blogs a lot about it. The guy that runs the site, have I been pwned? Help me out, Troy. Let me see if I can find it. Yeah, Troy someone. I don't know if it's listed here. If you go there and look at the about, it's a guy, what is it? Troy Hunt. And he's a long time, like Microsoft security guy. I would follow those two to start for sure. Yes, ma'am? Well, am I aware of any issues of running more than one security plugin in WordPress? Not specifically, other than to say, well, other than to say if you're using a plugin for anything. So let's say you're using multiple gallery plugins. There's always a chance, the more plugins you have, the more opportunity you have for conflict. Security plugins are typically a bit resource intensive for your server. So if you have two security plugins installed and they're doing some of the same stuff, then I wouldn't recommend it. But I don't know of any conflicts specifically. Anybody? Yes, sir? For initial WordPress core hardening, what do you miss? Boy, what is my, for initial WordPress core hardening, what is my pet peeve and something that gets missed? You know, it used to be the recommendation was to change the database table prefixes in WordPress. But a guy named Aaron Campbell, who you should also follow, by the way, he's the .org security team lead. He and I had a discussion last year about the unimportance of the standard advice for securing your site, like changing the database table prefixes, because his line of thinking is that they don't get in from your database, right? An automated script doesn't get there. If an automated script gets in through a week username or password, they're going to be able to list your database table prefix. So to answer your question, I think my pet peeve would be that there's so much of that that kind of old advice out there that really doesn't apply today that still gets put out there as nothing to do to secure your website. It really has gotten a bit simpler than that. It really comes down to blocking bad traffic and strong passwords at the two most important core recommendations that I would make. Anybody else? Yes, sir. What was the second part of the question? Do we need to use single installation for best for multiple sites, or you can use multi-site? So the question was, in terms of security, what's more secure, WordPress itself or WordPress multi-site? I think the answer to that in my experience is that both are secure, as secure as they can be, right? Because it used to be WordPress multi-site was a separate code base, but with WordPress 3.0 the multi-site features were built in to WordPress core and now you just activate them in order to get the extra administration areas and that sort of thing. So my answer to that question would be multi-site is as secure as WordPress single site is. Anybody else? Yes, sir. You're right. And the question is what about the human factor of security? Unfortunately, I don't have a very good answer for that because as you've just pointed out the human factor is a wild card, right? There will be people who just say I'm not going to use a password manager because that's going to be more difficult for me to figure out. I'm just going to write my stuff down and put it under the keyboard. So one maybe simple way and I should have mentioned it before is to utilize two-factor authentication. So if you can't remember your password or if you want to keep using the same password for other stuff implementing two-factor authentication would be a good step. And if you don't know what that is that means you go to your WordPress site to log in, but rather than seeing your login screen you're presented with a screen by some 2FA two-factor authentication provider that says enter your email and then you press a button and you get a text with a code. So it basically means you have to know this login, you have to have this device on you, you enter the code and then it lets you to your login screen. So that'd be my best piece of the device there. Yes, sir? I thought the quote against two-factor authentication has probably the worst idea that's come along in the webinar. As someone who spent my over a week without my WordPress.com account for two-factor authentication it's a very flawed solution. Okay. What happened was we moved from Ophel from the process I got to myself. Ah, yeah. I've been on WordPress.com for a long time. I went to log in and guess what? It sent an SMS to my phone in Ophel. My phone in Ophel was not long with me of course. Sure. So I thought, oh yeah, I'll just call the person who's calling and say my SMS is coming. Oh no, the phone was out of service. Yeah. So I called Rogers. Rogers said, oh no, we can't give you that phone number because yes it's out of service but it is in limbo right now while we wait to see if it lasts three months before the other person gets it back. So now I'm sitting waiting for three months because Rogers can't give me back my phone number and because WordPress doesn't answer emails or phone calls. Otherwise, I eventually went on to WordPress.com with another account and sent an email off to support people and got a phone call back from somebody who was able to sort the thing out and it took over a week. So if you do use 2FA, make sure you keep the little sheet of backup codes. I will keep the backup codes whilst I see if I move it was in a box somewhere in the basement and so I use those and don't rely on your phone. There you go. So that's one negative vote for 2FA but there's something else to keep in mind also with that story and that's a nightmare story. That is a real risk, right? But there are other 2FA providers too that might offer you a little easier communication path. There you go. That's another good tip. There you go. Well, thank you all for coming. I appreciate it. I'll be around the rest of the day. Thanks.


For five decades, architect and urban designer J. Turnbull has used his expertise in preservation and land development to implement pivotal preservation campaigns across the country and especially in his hometown, San Francisco. Turnbull gained early experience at Skidmore, Owings and Merrill, as well as in the New York City Office of Midtown Planning and Development. In 1974, he became the first staff architect at San Francisco Heritage, where he met Charles Hall Page. The two went on to develop the world-leading preservation and architecture firm, Page & Turnbull, which has found creative strategies to give old buildings new life for more than 40 years. Among the firm's many achievements is Splendid Survivors, a historic resource survey that documented the history, significance and relative value of several hundred of San Francisco's most notable buildings. This influential book was critical in the development of San Francisco's New Downtown Plan in 1985, which established revolutionary zoning laws that continue to protect the city's historic structures today. As former staff member and later president of the Board of San Francisco Heritage, Jay has touched nearly every important historic renovation project in San Francisco. Through Page & Turnbull, he helped turn the San Francisco Ferry Building into a bustling marketplace, office space and public gathering spot. While chair of the Mabek Foundation, Turnbull was instrumental in launching a $21 million program to rehabilitate the rotunda of the Palace of Fine Arts and its park, following damage from the 1989 Loma Prieta earthquake. Turnbull's thoughtful approach to preservation and adaptive use has earned him numerous state and national honors. He has been named a fellow with the American Institute of Architects, been a peer reviewer for the U.S. General Services Administration's Design Excellence Program, served as visiting critic at Yale University and the University of California, Berkeley, and was honored by the California Preservation Foundation. Through Page & Turnbull, Jay has also been involved in urban planning and landmark conservation projects spanning from Anchorage, Alaska to Charleston, South Carolina to the Grand Canyon. Turnbull's many successes exemplify his unwavering commitment to preservation, which has made a direct impact on hundreds of buildings, national parks, government agencies, heritage neighborhoods and entire cities across the country.


Live from Las Vegas, it's theCUBE! Covering AWS re-invent 2019. Brought to you by Amazon Web Services and Intel, along with its ecosystem partners. Welcome back to theCUBE, the leader in live tech coverage live from AWS re-invent 19 in Vegas. I'm Lisa Martin with John Furrier, and we're going to be taking something that was an exclusive from John's interview with Andy Jassy from a couple of days ago. Something that he told John, and we're going to be talking about it here with Verizon. Please welcome Steve Zabo, head of Global Products and Solutions IoT in 5G Edge. Welcome Steve. Thank you, thank you for having me. This is an exciting day. It is an exciting day. So one of the things that Andy Jassy told John in that exclusive interview that went viral, if you haven't read it, check it out, was that companies are going to want to eliminate network hops and find a way to have the compute and the storage much more local to the 5G network edge. Tell us, what did AWS and Verizon just unveiled this morning? Yeah, so today it's all about Verizon's network and AWS cloud, right? So we've taken what they're calling wave length. They're centralized cloud platform. We're moving it into Verizon's network, fully integrated. This is 18 months of engineering effort. So this isn't something that you just wake up and you have access to. This is a lot of blood, sweat, and tears that the companies have put in together to get this opportunity. What this does is it takes their cloud capabilities. It puts them on our network fully integrated with the radio access layer so that customers will have access to everything that they were using from an AWS perspective, but then also be able to leverage Verizon's network capabilities. So all the APIs, the currencies that Hans talked about on stage today, giving developers and businesses a like the opportunity to leverage the best of those and go ahead and leverage the bandwidth, the late C type use cases, and really transform the way that folks are thinking about leveraging the network. You know, Steve, one of the things in the network, in the computer industry, everyone always talks about trade-offs, hops on the network, spectrum. I got a longer range or shorter throughput. 5G's got some pretty significant bandwidth. Up to 10G's, gigs on that. That's phenomenal, but the footprint's a little bit different, so that begs the question for high bandwidth needs, whether it's gaming, immersive experiences, whatever. You got to bring the compute. This is the whole thesis of Amazon's shift. They're bringing Amazon to the edge. You guys are providing it. What's different about the Verizon 5G that makes this a unique opportunity? Is it the throughput? Is it the topology? Is it the... We like to think it's a little bit of everything, right? Tell us how it works. Yeah, I mean, listen, at the end of the day, we have 5G, it's opening up the eight currencies. When you factor in 5G edge, that's when you really see the power of 5G, and then when you layer on the AWS Wavelength stack integrated into the network, it just gives an opportunity for folks to take advantage of these eight currencies. 100 feet behind me at our booth, we have Bethesda Gaming, and that was one of the things that we talked about, but if you think about it, they have an Orion gaming platform. They leverage AWS today. They want to reach out and have the ability to have their gaming platform stream to Verizon customers using mobile devices. If you think about the fact that you can almost take the council out of the home, folks are literally leveraging GPU, CPU intensive graphic and gaming streaming content, and they're using a Bluetooth controller, and they're doing it on a Verizon 5G device. I mean, who would have thought that you'd be able to do that? And you can see it, and that's live in Chicago. Now they're piloting it on our network, so. Talk about the partnership with Amazon. You mentioned it wasn't just an overnight thing. Multiple months in the making, announced on the station, Wavelength was their product. That's the stack. It's essentially an outpost for telco. That's the way I'm calling it, whatever you want to it. There's some things in there, but they still got to deploy it. What does that look like? How long have you guys been working with Amazon? You share some details on the relationship. Where is it located? Is it in your network, close to the edge? How close is it? Has it all worked? Yeah, so we'll touch on what we can here, but it's live in Chicago, so that's our first market. We'll take an approach to announce it similar to what we've done with our 5G city announcements, which is we'll work with our partners, we'll talk internally, and then we'll announce those as it makes sense into other markets and cities. Currently, the way that it works is is at our SAP sites or our service access points, AWS will have their equipment. It'll be tightly integrated with our radio access network, which is when you can see the benefits of the low latency and the compute all kind of working together. The way for folks to procure that is they would go through, if you're an AWS customer today and you're getting storage and compute, you would be able to access that through AWS's portal environment. It'd just be labeled as Verizon 5G edge capabilities. If you're buying bandwidth, if you need pro services help, other network service capabilities, you'd work with Verizon just like you do today. It's a true partnership opportunity, and it allows us to kind of work together and kind of head on this journey. So the key thing is there, Amazon console access, flick, provision. You're in, yeah. We did all the hard work and engineering between the two of us to make it as easy as possible for the developers and the businesses quite honestly. We want what they're familiar with today, both on our network and the tools that they're using in the cloud to be the same experience that they have, only just with the benefits of wavelength and with Verizon 5G edge. Any feedback you can share on early returns or early engagements or early tinkering and playing around that you could share? You know, I would tell you that it's operating as we would expect it to, and that's why I would encourage people to go over to our booth and see what's happening over there because it's, when I say that it's live and it's working, it isn't a video, it isn't anything that folks are talking about. This is on our production network. Bethesda's actually gaming with it, leveraging AWS wavelength, and we've got other customers that we're working with as well. And if they're not on here on site can they go to the website? Is it online now or? Yeah, you'll be able to see whether you go through Verizon's web experiences or AWS portal, it will redirect you either way to learn more. So if you want to learn more about the capabilities on Verizon's side, you'll punch into our site. If you want to go learn more about wavelength and what Amazon's doing, we'll punch it back to them. So for, let's talk about benefits. You gave a great example of somebody gaming and that they're accessing live streaming content from wherever they are from the Bluetooth device. So I can understand it from that perspective. But from a business perspective, business apps to business apps, what are some of the projected benefits that enterprises are going to see with that respect? Well, I think a couple of things. One, it's going to open up use cases for latency intensive. So I brought up Bethesda for a reason. They're cloud gaming to actually stream Doom, which is the game that they're demoing in our booth. They couldn't do it without the edge, right? They would not have the real-time gaming capabilities to actually work without it. When you start thinking about retail environments and getting into AR, VR, these immersive experiences to get customers to come into the four walls of your retail building the ability to have application services that'll reach out and engage with consumers for a variety of things, whether that's helping them with their buying experience or just for the benefit of your business gathering intensive sensory data and kind of getting into the AI and ML of how your business is operating on a day-to-day basis. It opens up a variety of things. It's really an ecosystem, which is what I think the power of this partnership is all about, right? We're bringing our customers and our network and buying that with AWS services and their developer community. And I think it's a tool in the developer toolbox that whether you're a developer at a large enterprise, small business, public sector, et cetera, it's something that you can use your imagination to go out and do something with and kind of test the balance of three to three. Anything about the headroom available in terms of future proofing? You got optimization closer to the edge edge. You got inside the network capabilities to manage software, to manage resources, kind of a new architecture. A new way to think about resources allocation from bandwidth to compute the data. You bring up a great point because that's something you had mentioned earlier about the difference between what we've done versus Kolo or something like that. And this is a full integration. So the ability to architect something that did not actually exist before. WaveLength is new for AWS. Our 5G edge and our 5G network capabilities, integrating that seamlessly so that the developer and the enterprise business can have access to that with having a minimal impact to their user experience is really important. And then you figure on the layering of possibilities as they start getting more familiar with it. Andy Jassy mentioned on stage with your chairman and CEO comment. I don't know if he was just pre-ambelling on the intro, but he said, Verizon, the leader in 5G. I'm sure he meant that. For the folks that aren't following the 5G situation, are you guys the leader and what way are you leading compared to the others? How should consumers think about 5G? It's almost like this magic pixie desk. It's almost a magic. But wait a minute, we can't have those speeds. Some say hype. What is the reality of the 5G and why are you guys being called the leader? Yeah, well, we were the first to market with 5G. So by default, I think that makes us the leader. But we'll be in 30 cities by the end of the year. The fact that we're the first to have 5G mobile edge computing capabilities, it's integrated with AWS WaveLength. That's to my knowledge, not out there in the market yet today. The ability, the fact that we have this live in Chicago, we have customers using it. It's demonstrating real world use cases on a live production network. I mean, we're excited about it. Something we're proud of and it's something that we expect to watch grow and actually ideate with the customers in mind. Congratulations. Thank you. One of the things that Andy and John talked about and with this whole, not just the notion of transformation and there's a lot of talk about transformation today but also the fact that businesses, you know, the vast majority that are around today, if they're not already iterating and moving towards digitalization and modernization, that there are a million companies probably doing the same thing or very similar that are going to be able to take them over. But that's a hard change for a legacy enterprise to be able to do. This new ecosystem that Verizon and AWS are building and delivering, what do you guys see together as its ability to be an enabler to transform businesses? Such that, you know, we don't see a business doesn't go by way of, you know, Toys R Us, for example. Yeah. Well, I think the fact that 5G and the Edge, it offers you to touch out and reach the customers in a way that you couldn't before for your business. That's one. Two, this is geared around 5G and Edge and that's when you really see the power of what we're doing between Verizon and AWS. But one thing that I'd like to highlight is wherever you're at on your digital transformation, some people are going to be starting from zero and some are going to be more advanced. I mean, that's a reality of kind of the technology and business alike. We're actually have solutions today. AWS has products today. They're already in the cloud. We have LTE capabilities and other network services, capabilities, virtualized networks, software defined network capabilities. We can work with customers and help them kind of grow into where they want to be. We do not want somebody to feel like they're buying in and almost isolating themselves into a technology. What we're all about is helping them build the solution that's right for them at whatever point in the journey they're at and then helping them grow into where they can be with 5G and Edge compute. Yeah, I think this is also instructive for the industry structure. You look at the landscape of kind of everyone thinking about replatforming their business in the modern era. You guys have a great footprint, great leadership. Just the idea of this win-win that makes you guys so much more powerful for future applications. I mean, I can almost see the edge just becoming a very fertile ground for entrepreneurial activity, applications that you guys are going to be powering. I mean, born on the edge might be the new phrase. Not born in the cloud, born in the edge. You can trademark that. No, we're excited. I mean, listen, it could be anyone from two people in a garage developing something to developers at a small, medium, or large business taking advantage of use cases and things that might not been achievable. Local education. I mean, it's endless opportunity there. Yeah, opportunities in energy management and sustainability we're very proud of. Education, healthcare are going to be areas that we'll focus on. So there's a lot of opportunity out there. We're at the forefront in our opinion at helping kind of just jumpstart that ecosystem and we're excited about it. Really, really great. I'll echo that. Congratulations and thank you for sharing with John and me more detail about AWS and rising in this new ecosystem that opens up tremendous amount of opportunity. We appreciate your time, Steve. Thank you very much for the time and we're excited. It's a big day. It is a big day. Big announcement. For Steve and John Furrier, I'm Lisa Martin. You're watching theCUBE from ReInvit19 from Vegas. Thanks for watching.


Hello everyone, I am Dr. Sneha Chaudhary, second year post-graduate student in the Department of Rated Diagnosis, Maharshi Markandeshwar Institute, Mulana. The topic of my paper is role of MRI in evaluation of malignant lesions of oral cavity and tongue. Oral cancers are among the top three cancers in India. Among the malignant lesions, squamous or carcumus predominant, accounting for more than 90%. Most common risk factors include long-term overuse of alcohol and tobacco. Other risk factors are HPV, poor oral hygiene and positive family history of oral cancers. Five-year survival rates of oral cancer and oral and tongue cancer is approximately 50%. Cross-sectional imaging has become a cornerstone in pre-treatment evaluation of these cancers as it provides an accurate information about the extent and depth of the disease which helps in appropriate management strategy and indicate prognosis. CT and MRI are often complementary in assessment of oral cavity and oral pharyngeal pathologies. This is AGCC TNM classification of the oral cavity carcinomas. MRI readily reveals tumor invasion and spread to the surrounding structures due to its excellent soft tissue discrimination. So MRI can be used to assess the extent of local and regional tumor spread, the depth of invasion and the extent of lymph adenopathy. Also, MRI does not expose the patient to any harmful radiation and should also be used in the state of CT where dental fillings can obscure the region of interest. The sensitivity, specificity and accuracy of MRI in detection of mandibular invasion is also high and MRI can detect marrow invasion by the tumor involvement of the base of tongue, floor of mouth and extension into the oral pharynx. Ames and objectives are to evaluate the role of MRI in local regional staging of malignant lesions of oral cavity and tongue and to correlate these findings with clinical, surgical, anatomical and pathological findings. The study was carried out in the department of radio diagnosis in Maharshi Markandeshwar Institute. 30 cases were included in the study. The inclusion criteria was any histologically proven or clinically suspected malignant lesion of oral cavity and tongue and all the patients with contraindications to MRI were excluded from the study. The study was carried out on the 1.5 Tesla MRI machine and the MRI protocol is stated. The results and observation of my study was that the cancer was predominantly found in males while on site of involvement followed by buckle mucosa and alveolar followed by heart palate. The age incidence of my study showed that most of the patients were between 30 to 60 years of age group and my study showed a moderate agreement between the clinical and MRI T staging and a fair agreement between MRI and clinical N staging. A good agreement was seen between MRI and histopathological T and N staging while a poor agreement was seen between clinical and histopathological T as well as N staging. So the first case is of a female who presented with a non-healing ulcer on the left lateral part of the tongue which bleated on touch and histopathological studies revealed a moderately differentiated squamousal carcinoma. These are the MRI images which shows a large infiltrating mass involving anterior 2 third of the tongue and the base of the tongue appearing heterogeneously hypo intense on T1 weighted images whereas it appears heterogeneously hyper intense on T2 weighted images with a contralateral extension on the right side infiltrating into the floor of mouth involving glossopharyngeal circus, left tonsil, velicular, left paraphernal space and mastigator space with higher bone erosions and nodal deposits. This is the second case of a 67 year old male patient who presented with the ulcer on the right side of the tongue since two years he was a chronic smoker and of tobacco chewer. The histopathological reports were consistent with poorly differentiated squamousal carcinoma. The MRI images revealed an irregular ulcerated mass lesion which appeared hypo intense on T1 weighted images and heterogeneously hyper intense on T2 weighted images in the right little border of the tongue no involvement of the extrinsic muscles or any contralateral extension of bone erosions were seen. In this next case a 32 year old male presented with a complaint or difficulty in opening the mouth and pain in the left side of the jaw since three months and MRI images revealed a large heterogeneously infiltrating mass showing epicenter along the left bachal mucosa which appeared iso intense through the tongue muscles on T1 weighted images and hyper intense on T2 weighted images in extending till the left paraphernal space with involvement of the mastigator space and erosions of the left premise of the mandible. In the next case a female presented with a history of pain burning sensation past discharge bleeding with the ulcerative lesion in the right side of the alveolar ridge since four months. On the MRI images we can see the structure of the right maxillary alveolar arch by a large soft tissue lesion appearing iso intense through the tongue muscles on T1 weighted images and mildly hyper intense on T2 weighted images with associated destruction of the right entry floor. Intrasonous extension was also seen. Next case a patient presented with a complaint of pain and swelling over the right side of the cheek the patient was a chronic smoker and histopathology revealed a poorly differentiated squamous cell carcinoma. MRI images showed a large ulcerative proliferative soft tissue lesion centered in the right bachal mucosa appearing hypo intense on T1 weighted images and hyper intense on T2 weighted images with extensive destruction of the right half of the mandible with fracture and cranial deviation of the angle of the mandible. Last defect was seen along the right cheek with fistulous communication with the oral cavity. So to conclude my paper I would like to say that MRI showed a high correlation with histopathology for the thickness of the mucosal epithelium and both depth and width of the tumor. Preoperative estimation of the thickness of tumor and prediction of occult cervical nodal metastasis was satisfactory. The mucosal epithelium lamina propria and muscles of the tongue were clearly identified on MRI. New improvements in the MRI such as surface coil technology motion and flow compensation imaging strategies has made MRI as the modality of choice for assessment of oropharyngeal mouth and tongue soft tissue masses. TNM classification helps in planning treatment options. These are the references for my study. Thank you.


In our last lecture, we have defined power method for calculating Eigen vector associated with dominant Eigen value. Then, if the matrix is invertible and if the Eigen value which has got lowest modulus, so modulus of lambda n, if it is strictly less than modulus of the remaining Eigen values, then we can apply power method to the A inverse and then obtain approximation to Eigen vector associated with the least Eigen value. So, thus we can approximate the Eigen vector associated with the largest Eigen value in modulus or the least Eigen value in modulus. Now, what about some intermediate Eigen value? So, for that we need to have some approximation to say l th Eigen value and then we can extend our power method. So, that is known as inverse power method which we are going to consider today. After defining inverse power method, we are going to consider what is known as q r decomposition of a matrix. We have considered various decompositions of matrix, l u decomposition, then Cholesky decomposition and today we are going to consider decomposition of A into q into r, where q is going to be orthogonal matrix and r is going to be upper triangular matrix. Once we consider this q r decomposition, we can define q r method for finding Eigen values of matrix and then we are going to consider relation between q r decomposition of matrix and Gram-Schmidt orthonormalization process of the column vector. So, this is roughly the plan of today's lecture. So, let us look at the power method, recall it. So, our A is n by n matrix, our assumption is that Eigen vectors u 1, u 2, u n they form a basis for C n. Eigen values lambda j's are arranged in this manner and assumption is mod lambda 1 bigger than mod lambda 2 bigger than or equal to mod lambda n. So, these are our assumptions, not all matrices will have a basis of Eigen vectors, but then we have seen that if the matrix is normal, then there is a basis of Eigen vectors. If our matrix A has n distinct Eigen values, then also it has a basis of Eigen vectors. So, this restriction of Eigen vectors forming a basis is not too restrictive. There is a big class of matrices for which it will be satisfied. So, thus our assumption is A should have a basis of Eigen vectors and the basis of Eigen vectors, and then it should have a dominant Eigen value. That means, modulus of lambda 1 bigger than mod lambda 2 bigger than or equal to mod lambda 3 bigger than or equal to mod lambda n. So, afterwards we can have greater than or equal to, but the Eigen value which is biggest in modulus should be a simple Eigen value and it should be strictly bigger than moduli of the remaining Eigen values. Then we start with a non-zero vector.  So, alpha 1 is not equal to 0. So, with this assumptions and lambda 1 to be bigger than 0, that means lambda 1 is real and positive. So, we formed a raise to k z divided by norm of a raise to k z. Under these assumptions, a raise to k z upon norm of a raise to k z converges to alpha 1 u 1 by norm of alpha 1 u 1. So, this is going to be unit Eigen vector associated with Eigen value lambda 1. So, it is easy to calculate these iterates. You start with a arbitrary vector z not equal to 0, find these iterates. If your matrix A is a sparse matrix that means lot of zeros, then this will not be computationally too expensive and this is going to converge to a unit Eigen vector corresponding to the Eigen value lambda 1. So, this is the power method. Now, we want to look at extension of this method which will allow us to find Eigen vector associated with intermediate Eigen value. So, again our assumption is that A has n linearly independent Eigen vectors. A u j is equal to lambda j u j, j is equal to 1 2 up to n and mu is given to be an approximation to lambda l. So, some l th Eigen value in between it need not be the largest Eigen value or the least Eigen value in modulus. So, since mu is approximation to lambda l, its distance from lambda l is going to be smaller than distance of mu from the remaining Eigen values. So, mod mu minus lambda l is less than mod of mu minus lambda j, j not equal to l. Now, our Eigen values these are roots of characteristic polynomial. Now, for the characteristic polynomial, we can find an interval in which our Eigen value lies. For example, we had considered bisection method. So, we have got a polynomial. Now, you find two numbers or two real numbers where the polynomial has opposite sign. Then that interval is going to contain a root of the polynomial. So, you go on saying sub dividing it, getting smaller and smaller interval. So, in such a manner or by some other method, we can have a approximation to Eigen value lambda l. So, starting point of our inverse power method is you are given mu an approximation to l th Eigen value and our aim is to find approximation to an Eigen vector corresponding to this l th Eigen value lambda l. So, we have a u j is equal to lambda j u j, j is equal to 1 to up to n, mu being an approximation to lambda l, modulus of mu minus lambda l is less than modulus of mu minus lambda j, j not equal to l.  Assume that mu is not equal to lambda l. That means, it is not an Eigen value. So, a minus mu i will be invertible and then from this relation we obtain a minus mu i inverse u j to be equal to 1 upon lambda j minus mu u j. I apply a minus mu i throughout, a minus mu i inverse throughout. So, left hand side will be u j is equal to lambda j minus mu, a minus mu i inverse u j and take lambda j minus mu on the other side. Now, look at this. We have got modulus of mu minus lambda l to be less than modulus of mu minus lambda j, j not equal to l. So, if I take the reciprocal 1 upon mod of mu minus lambda l will be strictly bigger than 1 upon modulus of mu minus lambda j, j not equal to l, which will mean that 1 upon lambda l minus mu. This is going to be a dominant Eigen value of matrix a minus mu i inverse. So, the idea is now to apply our power method to this a minus mu i inverse. In fact, this is a minus mu i inverse. In fact, if we apply our power method, we need a dominant Eigen value.   So, here is our iterates. Choose z not equal to 0 vector and arbitrary vector define z 0 to be equal to z upon norm z and then z k. The k th iterate will be a minus mu i inverse z k minus 1 divided by its norm. So, it is exactly the power method, which is applied to the matrix a minus mu i inverse. Then this z k is going to converge to say vector w. This vector w will be Eigen vector associated with matrix a minus mu i inverse and 1 upon lambda l minus mu being the Eigen value. Now, what we are interested in is Eigen values of matrix a and associated Eigen vectors. Now, what we are interested in is Eigen vectors. We have obtained approximation to an Eigen vector w or we have obtained approximation to an Eigen vector w, which is Eigen vector of a minus mu i inverse. But this w is also going to be Eigen vector of a associated with Eigen value lambda l. So, a minus mu i inverse w is equal to 1 upon lambda l minus w. That will mean that a minus mu i w is equal to lambda l minus mu into w. So, this mu w will get cancelled and then you are left with a w is equal to lambda l w. So, w is Eigen vector associated with lambda l. So, this is the Eigen vector associated with lambda l. So, thus the iterates when we define in the inverse power method, they involve matrix a minus mu i inverse. You get an approximation and you get a limiting vector is w. This w is going to be Eigen vector of a associated with Eigen value lambda l. So, if lambda 1 is the dominant Eigen value power method, in that case we obtained Eigen vector associated with lambda 1. When you apply power method to matrix a inverse, we obtained approximation to Eigen vector associated with lambda n. And if mu an approximation to Eigen value lambda l is available, then we obtain approximation to Eigen vector associated with lambda l. Where lambda l can be, it is an intermediate Eigen value. You need to have approximation mu available. Now, here there can be some problem or we want to see whether there is going to be a problem. Now, what is the problem? We are saying that mu is going to be an approximation to lambda l. Lambda l is Eigen value. So, our a minus lambda l into i, it is not an invertible matrix. So, now, when you are going to have mu to be a better and better approximation to your Eigen value lambda l, a minus mu i inverse can be ill conditioned. And then whether this is going to pose a problem. Now, we will see that it does not matter in this particular case. So, we have in the inverse power method, we need to find z k to be equal to a minus mu i inverse z k minus mu i inverse.  Now, we are not going to calculate a minus mu i inverse.  We want to calculate this. So, I am denoting it by r k minus 1.  It is the coming from the iteration step. So, we need to calculate this r k minus 1 and then divide by its norm. So, that you get the next iterate. So, now what I was saying that whether we are going to have some problem about the stability. So, here if a is a invertible matrix and lambda and mu are eigenvalues of a, then modulus of lambda is less than or equal to norm of a. It follows from our basic inequality. You have got suppose you have a x is equal to lambda x, x not equal to 0 vector. Then, take norm of both the sides. So, norm of a x is equal to norm of lambda x. This will be modulus of lambda times norm x by property of norm. Hence, mod lambda will be equal to norm a x divided by norm x and this will be less than or equal to norm of a, where norm of a is maximum of norm a z divided by norm z, z not equal to norm z. So, this will be equal to 0 vector and thus we have got mod lambda to be less than or equal to norm of a. If you consider a y is equal to mu y, another vector, another eigenvalue mu and associated eigenvector. If a is invertible, a inverse y will be equal to 1 upon mu times mu times y. Hence, using again this similar inequality, you get modulus of 1 by mu to be less than or equal to norm a inverse. So, the condition number norm a into norm a inverse will be bigger than or equal to mod lambda by mod mu, where lambda and mu are any eigenvalues of matrix a. So, here we have a is invertible matrix, lambda and mu are eigenvalues of a, mod lambda to be less than or equal to norm a, 1 upon mu, mod mu will be less than or equal to norm of a inverse. So, take the quotient, this is the condition number of matrix a, it is bigger than or equal to this and if you consider eigenvalues are arranged in descending order of modulus, then we have mod lambda 1 by mod lambda n less than or equal to norm a into norm a inverse.  So, norm of a minus mu i inverse is going to be bigger than or equal to 1 upon modulus of mu minus lambda l, if mu is approximation to lambda, a good approximation.  Now, we have considered the perturbation theory, in the perturbation theory if your matrix a is ill conditioned, that means if norm a into norm a inverse is big, then the solution is sensitive to the perturbation in the right hand side. We had looked at a x is equal to b, perturb b slightly. So, consider the nearby system a of x plus delta x is equal to b plus delta b, then even though norm delta b by norm b is small, norm delta x by norm x can be big. In fact, we had proved this inequality, norm delta x by norm x is less than or equal to norm a into norm a inverse into norm delta b by norm b. So, this can be small, but if you are multiplying by a big number, then norm delta x by norm x which is the relative error in the computed solution, x is the exact solution because of finite precision, instead of b it is going to be b plus delta b. So, x plus delta x will be computed solution, the relative error in the computed solution will be norm delta x by norm x. So, this can be big even though norm delta b by norm b is small. Now, this situation is going to occur in our case, we are calculating our z k. So, the k th iterate was given by a minus mu i inverse z k minus 1, the earlier iterate divided by its norm. So, we need to calculate a minus mu i inverse z k minus 1, which we denoted it by r k minus 1. So, that means we need to solve a minus mu i r k minus 1 is equal to z k minus 1.  This is what in general like at each stage we will be solving a minus mu i r is equal to cube. Now, in practice instead of this you are going to solve a minus mu i r cap is equal to q cap, where q cap is a nearby vector. So, even though norm of q minus q cap divided by norm cube, even though this is small, because a minus mu i inverse has a big norm, norm of r minus r cap by norm r can be big. Now, this is something we need to worry about, because as such when we talk about the approximation mu should be available to lambda l, then the better is the approximation our task should be simplified. It should not face such difficulties that when you have better and better approximation of mu to lambda l available, then your problem becomes more and more ill conditioned. So, then the relative error in the computed solution becomes bigger and bigger. Now, in this particular application it does not matter, because we are not interested in the computed solution, but we are interested in direction. We are trying to find an eigen vector. So, what is important is the eigen direction. So, let me make it more specific. So, suppose our q the right hand side q. So, we are looking at q to be a minus mu i r is equal to q. So, we are looking at q to be a minus mu i r and this is the instead we are going to solve this.  So, q and q cap are near. So, the distance between c 1 and c 1 cap c 2 and c 2 cap that is going to be small.    So, norm of r minus r cap will be approximately equal to modulus of c l minus c l cap upon mod lambda l minus mu. Now, this can be big, but we are not interested in what r cap is, we are interested only in the eigenvector that is the direction. So, even when you calculate r cap then what we do is you normalize it. So, once the exact solution is r then you consider r upon norm r because when you are applying inverse power method you calculate a minus mu i inverse z k minus 1 and divide by its norm.  So, you are dividing by its norm. Now, instead of r k minus 1 you are going to have some r cap k minus 1, where if you consider the values they may differ, but for both r k minus 1 and r cap k minus 1 significant part is going to be in the diagram. So, this is the direction of u l and we are dividing by norm. So, then the numerical value they do not really matter. What is happening is the direction for both the exact solution and the computed solution. The component in the direction of u l becomes significant and that is why there is no contradiction in our method that when you have better and better approximation to lambda l that will give us faster convergence to eigenvector corresponding to eigenvalue lambda l. So, so far we have been talking about approximation to eigenvector. In the power method we had approximation to the eigenvector associated with lambda 1. When you apply it to a inverse it is eigenvector associated with lambda n and in the inverse power method it is eigenvector associated with eigenvalue lambda l. So, I have obtained an approximation to eigenvector. Now, what about eigenvalue? When we talked about exact eigenvalue and exact eigenvector we said that if I know eigenvector then finding eigenvalue is easy. You have to just find constant of proportionality. If v is eigenvector of matrix A then look at two vectors A into v and v. These two they are proportional. So, I find what is the constant of proportionality between the two. Now, here our eigenvector is only approximate. So, what best eigenvalue approximation can I choose? That means suppose I give you a eigenvector approximation and I want to know this is eigenvector corresponding to which eigenvalue. So, the best way it can be done is by considering the Rayleigh quotient. So, the Rayleigh quotient has got some minimization property which we now consider. So, you have q to be an approximate eigenvector and the question is what should be chosen as an approximate eigenvalue. So, the answer is choose Rayleigh quotient which is q star A q divided by q star q which is equal to eta. Now, let me tell you what is the minimization property of Rayleigh quotient. So, we consider eta to be equal to q star A q. So, our q is approximate eigenvector then eta is q star A q divided by q star q.  So, it is minus eta q star q. So, using this result this is going to be equal to 0 that means A q minus eta q is going to be perpendicular to vector q. Now, let z be any complex number and look at norm of A q minus z q it is two norm. Let me take it square. This will be equal to norm of A q minus eta q plus eta q minus z q. So, add and subtract here is one vector here is another vector our eta and z these are complex numbers. So, we have got vector eta minus z into q A q minus eta q is perpendicular to q. So, it is perpendicular to any multiple and which will mean that these two vectors they are perpendicular. So, we can use Pythagoras theorem. So, we have got norm of A q minus z q square A q minus eta q plus eta minus z q for any z belonging to C. Using orthogonality property this vector is perpendicular to this vector eta minus z into q.  So, what does this relation tells us? It tells you that two norm of A q minus z q is going to be bigger than or equal to two norm of A minus eta into q. So, our vector q is perpendicular to this approximate again vector. So, we cannot hope to find a complex number lambda such that A q is equal to lambda q, but what we are saying is look at the Rayleigh quotient q star A q divided by q star q that we denote by eta. So, now consider two norm of A q minus eta q. This norm will be less than or equal to norm of A q minus z q where z is any complex number. So, the Rayleigh quotient eta minimizes two norm of A minus z i into q where z varies over the complex plane and that is why if you are given q to be an approximate Eigen vector, the best you can do is choose Eigen value approximation to be the Rayleigh quotient q star A q divided by q star q.  So, that means lambda times q star q because lambda is a complex number. So, you have eta to be equal to in the numerator lambda times q star q divided by denominator is q star q. So, you get eta is equal to lambda. So, the Rayleigh quotient associated with the exact Eigen vector is nothing but the Eigen value itself and if it is not an Eigen vector then it minimizes two norm of vector A minus z i into q. So, now next I want to consider q r decomposition. So, the reason I am going to consider q r decomposition is I want to describe what is the q r method for calculating Eigen values of a matrix A or finding approximations to Eigen values of matrix A. Now, the description of q r method is easy. You will see that we can quickly describe what is q r method. What is difficult is to show its convergence and also to show its convergence and also to show its convergence by it works. Now, unfortunately these things are involved. So, we are not going to do these things in detail, but I want to tell you what is a q r method which is the currently used method for or it is the best method available for calculating Eigen values of matrix A. There is a relation between power method and q r method. What we have done is we have considered power method to find approximation to one Eigen vector. Now, instead of one Eigen vector you can consider several Eigen vectors together. So, that gives rise to what is known as simultaneous iteration and the best implementation of simultaneous iteration is done by q r method. So, anyway let us first look at what is q r decomposition of a matrix A. For simplicity let me assume A to be a invertible matrix and also let us look at the matrix to be real matrix. q r decomposition is available for complex matrices also, but it is just for the sake of simplicity. Let us restrict ourselves to real matrices. So, our assumption is A is real n by n matrix and we want to write it as q into r, where q is a orthogonal matrix that means q transpose q is equal to identity and r is an upper triangular matrix. So, here is A n by n real invertible matrix. M is to write A is equal to q into r, where q is orthogonal that means q transpose q is equal to q q transpose is equal to identity and r is a upper triangular. Let me write it as q 1 q 2 q n. So, these are the columns of q, q is going to be n by n matrix and these are q 1 is the first column q 2 is the second column and q n is n th column. Now, what will be q transpose? So, the q transpose will be given by q 1 transpose q 2 transpose q n transpose. So, q n is n by 1 vector, when I take its transpose then it becomes a row vector. So, here q 1 is the first column of q, q 1 transpose is going to be first row of q transpose, then q 2 transpose and q n transpose multiplied by q 1 q 2 q n. So, when I consider q transpose q it will be first row into first column first row into second column and so on the usual matrix multiplication. So, the i j th element is of q transpose q will be given by i th row here. So, that will be q i transpose multiplied by j th column here. So, it is q i transpose q j and our notation is inner product of q j with q i, q transpose q is equal to identity. So, identity matrix means one along the diagonal and 0 elsewhere, q j q i is the i j th entry of q transpose q. So, q transpose q is equal to identity is equivalent to saying that inner product of q j with q i will be 1, if i is equal to j, 0, if i not equal to j, which will mean that the columns of q are orthonormal. We had seen similar relation for unitary matrices, when we considered q star q, q is equal to identity. So, q star means conjugate transpose. So, q unitary that means the columns of q are orthonormal, it is exactly it is a special case actually that q transpose q is equal to identity that means the columns of q are orthonormal. Now, orthogonal matrix also satisfies q q transpose is equal to identity. So, from that we can deduce that the rows of q they are going to be orthonormal. So, orthogonal matrix is that matrix, which has if you look at any column, its Euclidean norm is going to be equal to 1 and that column will be perpendicular to any other column. And similar property holds for rows of matrix q. So, we have got q transpose q is equal to identity. So, that means inner product of q j with q i is 1, if i is equal to j, 0, if i not equal to j. So, the columns of q are orthonormal and q q transpose is equal to identity that means the rows of q are orthonormal. Now, we are trying to write a as q into r where q is orthogonal and r is upper triangular. So, let me write columns of a as c 1 c 2 c n. So, c 1 c 2 c n is equal to q 1 q 2 q n and then here is upper triangular matrix so, below diagonal all the entries they are going to be 0. So, now, let me equate the columns. So, the first column c 1 will be nothing but r 1 1 times q 1, second column c 2 will be given by r 1 2 into q 1 plus r 2 2 into q 2 etcetera. So, this is just the first column of matrix multiplication. So, we have got c 1 c 2 c n to be columns of our original matrix a. Suppose we can write it as q into r, then first column of a which is c 1 will be r 1 1 times q 1.  The second column c 2 will be r 1 2 times q 1 plus r 2 2 times q 2. Now, look at the first relation c 1 is equal to r 1 1 times q 1. Now, q 1 just now we have seen that the columns of q they form an orthonormal set that means, Euclidean norm of q 1 is going to be equal to 1. So, we have we have got c 1 which is the first column of a. This is r 1 1 times q 1. So, norm of c 1 its 2 norm is equal to norm of r 1 1 q 1. So, this will be modulus of r 1 1 and then norm of q 1 2 norm and this is equal to 1. So, for modulus of r 1 1 we have got choice. You can either choose r 1 1 to be equal to norm of c 1 or r 1 1 is equal to minus norm of c 1 2. So, you get q 1 to be equal to c 1 divided by norm of c 1 2 if you choose this. So, the matrix A is given to us that is the norm of c 1. So, the norm of c 1 is given to us. So, the norm of c 1 2 is given to us that means, I know what is its column c 1. I am trying to write it as q into r, where q is orthonormal r is upper triangular. So, if you want to consider q 1 the first column it will be nothing, but the c 1 normalized like the first column of a which we are denoting by c 1. It may not have Euclidean norm to be equal to 1. So, you divide by its norm. So, this looking at c 1 is equal to r 1 1 into q 1 we get q 1 will be nothing, but. So, first of all for r 1 1 we have to make a choice either you choose it to be positive or you choose it to be negative. So, for the sake of definiteness let us choose r 1 1 to be bigger than 0. Then our r 1 1 will be nothing, but c 1 divided by its Euclidean norm. Now, that is for the first. So, that means, we have determined what should be the first column of q and what should be the entry r 1 1 in the upper triangular matrix. Now, in the second one you have the relation c 2 is equal to r 1 2 q 1 plus r 2 2 q 2 we have already determined q 1.  So, now, I need to determine what is r 1 2 what is r 2 2 and what is q 2 q 1 is determined. So, these are the three things to be determined. I make use of the fact that q 1 and q 2 are perpendicular to each other. So, inner product of c 2 with q 1 will be r 1 2 and inner product of c 2 with q 1 will be r 1 2 and inner product of q 1 with q 1. So, this is going to be equal to 1. So, this determines r 1 2 because c 2 is known q 1 is known take their inner product and that will be r 1 2. Now, after this go to this relation and then take norm of c 2 minus r 1 2 q 1 its two norm will be modulus of r 2 2 because norm q 2 is equal to r 1 2 q 1. So, once again choose r 2 2 to be bigger than 0.  So, q 2 will be c 2 minus r 1 2 divided by r 2 2. So, in this manner we can determine the matrix q and matrix r. So, any invertible matrix can be written as product of q into r. So, in our next lecture we will consider the relation between a is equal to q into r and Gram-Schmidt orthonormalization process. Then I will describe what is the q r method and then we will consider an efficient way of finding q r decomposition matrix q r decomposition of a matrix by using what are known as reflectors. So, this we are going to do in the next lecture. So, thank you.