Alexander says, hello. Does anyone know of a way to save or export chats in Teams? Yes. There are PowerShell commands and third-party tools that will do that. I know ShareGate will do it. You can actually migrate it and there's an export function where you can basically migrate it and then export it out in Excel. I know there's a PowerShell command for it, to be able to grab it and export it. There's also, it depends if you're coming from the, are you coming from just a standard in-user perspective or are you coming from technical? Because if it's just an end user in terms of saving the chat, yes, you can actually just do the ellipsis button right next to the chat and save it and put it in your save folder so you can come back to it later. So there's that. There are two. Yeah. Then the next one is, you can also click on the ellipsis button and email it. If you're trying to send a chat, if it's in conversations in a channel, are we talking channel or are we talking private chat? So you can just email it out. So are we talking about the individual or are we talking about I need to for the whole team or a whole forensic discovery because it's for legal reasons? Yeah, we've got two sides to- That's so funny, Christy, because I see it, it's so funny. Coming it from two perspective is I immediately see it as an admin or a compliance person. We need to get access to chats and export them. You immediately see it as an end user, if I want to get my own chat. So I mean, it's a good point to think like, well, whose chats do you want? How many of the chats do you want? Where do you want them to end up? What permissions do you have to be able to do this? Are you an owner? Can you do that if you're an owner? Yeah, because that's a question I get all the time. Because it's from an end user perspective, I deal with end users, so it's a pretty common question for me. The one other point of clarification too, just one other point of clarification too, is that because some people sometimes refer to the threaded discussion in a channel as the chat, that is not a chat. And it no longer says chat, it said conversation before and now it says post. But in chats, it still says chat. Yeah. So two different things, there's different technology under underlying technology there. That's right. So yeah. And the one other thing I would add too, is that I was going to jokingly, one of my not famous non-answers for the question was that, you invite more people into the chat and then it will live on because everyone then has access to the ability to save that. So that could be your backup, your archival, is that invite a bunch of other people to the chat, then they all have that whole chat history and it's dispersed across multiple locations. Life hacks and Christian. Life hacks, all the important ways of doing things. That's right. Yeah, yeah. So the chats just keep disappearing with what keeps happening. I keep putting my hands over my eyes. Also not helpful. I think it's getting removed, I don't know why. To Christie's point, right, how much are they trying to capture? Is it just that one subject and whatever they can email or hey, how about just highlight, scroll down and put it in one note, copy and paste. That's another option too. You don't even really need to, because in Outlook, in Outlook you can go, you can go to your folders in Outlook. Sure, it's in Drive Heaps, wait a second. Sorry, sorry, Christian. No, no, no, I was just saying, because in Outlook, if it's chat chats, your chats are actually saved in the conversation folder inside Outlook. So, you know, there's that too, from an individual, like, so yeah, I mean, you've got the ability to be able to pull it. Does it still do that? Because I don't see that anymore. Yeah, it's still there. Yes, yes. Now I'm looking. If you wanna be super fancy in the Compliance Center, you can actually do some pretty cool things depending on the license level you have to where you can actually pull it. But I mean, obviously that's gonna require some level of Compliance Administrator role, but there's some pretty neat things you can do. Yeah, and look, and who's pulling it too? Because you gotta be really careful.  Why? Why? Why are you trying to do other police investigating and a report of theft or, you know, why? You know, you've gotta have a good reason to go and stop exporting and pulling chats. No reason, no reason. I see it all the time. Oh, just because we wanna back up everyone's chats, it's like, why? Because Link and Skype for Business used to have the conversation history, and I'm looking at mine right now, and I do not have that. So I wonder if it's something that has to be turned on in order for it to capture it, or I'm in two different tenants, looking at it, and neither one has that. So I wonder if it has to be turned on. Because I missed that, because that was helpful to me. I see mine. Okay. This is going to be, probably be a tip Tuesday. Sherry's like, please hold, I'm writing this down. I just did, in my handy-dandy rocket book. Cool.


Hey YouTubers, it's Deidre, owner of Spalding Decon Plano. I was not going to bother with a video of this job, but now that I'm here, I'm like, mmm, this one's a good one. We have a quarter to tell.


Hi, folks. I'm Jordan, and I maintain NVM, which is a widely used Node version manager. There's lots of reasons a version manager for Node is useful. If you're offering packages, it's important to be able to locally test in different Node versions. And if you're running an app and preparing to upgrade from, say, Node 10 to Node 12, it's really helpful to be able to quickly switch between them so you can ensure that the upgrade will be as low risk as possible. For most users, installing NVM is a one line command found in the projects readme. If you have any trouble, please file an issue and fill out the full template, and I'll do my best to help. After installing NVM, the first thing you'll want to do is install a Node version. So we have to decide which one. I'd recommend starting with the latest version. So for that, you'd type nvm install node. This will download and install a binary or compile from the source if needed, and then we'll use it, which you can verify with nvm current. NVM will also cache the downloaded archive so you don't have to download it more than once. The first time that you install a Node version, NVM will set it as the default alias, which means that it will be auto-used when the shell starts. You can check this with nvm alias, or nvm ls, which includes the alias listing. You'll see the default pointing to version 14.4.0, and n slash a are not applicable for most of the others. All the yellow ones represent LTS release lines, which stands for long-term support. You can use LTS slash star, LTS slash erbium, et cetera, as a shortcut to install that LTS version. So now let's install some more Node versions. I'm going to type nvm install LTS slash argon, which installs version 4.9.1, the last supported version in the argon release line, which is now end of life. And with nvm ls, we can verify this. And as you can see, version 4.9.1 is installed. The arrow means that's the current version. And the LTS argon alias points to it down at the bottom. So you can get the latest LTS version with nvm install dash dash LTS, which at this time is version 12.18.0. And with nvm ls, we can see that that's installed. The arrow means it's current because it's been used. It's LTS slash erbium. And LTS slash star also points to the latest one, which at the moment, the latest installed one, which at the moment is that. Now that we have three versions installed, we can switch between them with nvm use. So in this case, I typed 14. You can type 14.4 or 14.4.0. You can also use LTS slash argon or any of the other aliases. The custom aliases are supported as well. And nvm use dash dash LTS is a shortcut for LTS slash star, which is the same thing. And these all use your locally installed versions, not the remote ones, only when installing. So this is a check what's available. And for any commands, you can check nvm dash dash help to see what's available with hopefully a helpful explanation. And if you have any questions, please feel free to file an issue and ask. Last October, nvm became the first project to join the newly merged OpenJS Foundation. The primary goals for being part of the foundation are to ensure that the project survives long term and can grow beyond its current state, which is having a single maintainer. Here is a few things that I'm focusing on for nvm over the next year. The main one is speed improvements. nvm invokes npm when you're using a version. And this is quite slow for some users. This needs to be fixed, but without bringing back the bugs that invoking npm prevents. The next thing is official support for release candidates and nightly builds of Node. This is already possible with a few hacks, but I want it to be properly supported. The next one is the ability for nvm to automatically update itself. So you don't have to keep looking up the install script to grab the new version number. And then the last big one is testing. The current tests that nvm uses to validate itself are cumbersome to write and, more importantly, to read. And we need a better solution. So you might be asking yourself, how can I help? The big one, of course, is contributing. You can find an issue with the pull request wanted label and gonettes. But remember, the most important kind of contributions are not code. They're making documentation more intuitive and approachable. They're reproducing reported issues and distilling reports into failing test cases, things like that. And if you don't have the time to contribute, that's fine. There's also sponsorship available via GitHub. That helps as well. Thanks, everybody. And look forward to hearing about all the great ways you're all using nvm.


Welcome back to the channel, everybody. For those of you who are new around here, my name is Michael, AKA Dr. Chalini, and I am a diagnostic and interventional radiologist in New York City. On today's video, we are going to be reacting to a video from one of my favorite people, Dave Ramsey, who is a straight shooter. I watched this guy pretty frequently. He has some pretty good tips on financial success and financial freedom. The video that caught my eye today and the reason I'm reacting to it is because the title of this video is, I'm a multimillionaire doctor unmotivated to work another 13 years. I don't know why it's 13 years, but I guess we'll find out. I have not seen this video, so let's go ahead and get into it and see what it's all about. Good afternoon, I'm Dave. Thank you for taking my call. Sure, what's up? I'm 37 years old and I need help finding motivation to work for another 13 years until I'm 60 years old. I have about 2.1 million in various 401ks and I have no debt. My three houses are paid off and I've paid off my medical school loans. I mean, this is impressive. This is very impressive, especially for a doctor because there's a lot of doctors out there who due to their salary increase, they just overspend and match their increased salary and that's exactly what you're not supposed to do. And it sounds like this woman did everything right. I'm about 1.6 million in stocks and bonds and a 20K savings emergency fund in my bank. As I said, I'm 47 years old. I've never married, no kids. I support my parents who are in their 80s. I make about 200K a year as a doctor. And that's actually on the low end of a doctor. So she's probably a pediatrician or family medicine or internal medicine doctor. I met with a financial advisor last year and she said I could stop working and have enough money to support myself. What do you like about your work? What do you not like about it? Ooh, I'm curious to see the answer to this. She's pretty much financially set up for retirement. It's very strange to me that she wouldn't want to just work another 13 years. I bet it's burnout. And I haven't seen this video, but she's probably just burnt out. What do I not like? Don't work is overwhelming these days in the setting of COVID. I always work more than my paid hours and I accept that. If I truly didn't feel the need to work, I might go to music school. I play the piano and the violin and it's always been a dream of mine to apply to. All right, so we're missing something here. There's something she hates about work because obviously she loves taking care of patients. That's what we all love to do as doctors. That's why we do what we do, working during COVID as a physician. The PTSD has been real. A lot of physicians have talked about it. I saw a really good article from a physician, I believe at Harvard who posted about PTSD from COVID and how it's affected her working going forward. It's been quite a stressful year for every doctor, but I'm curious to see something we're missing here.  Is there something? This is what I've talked about in my prior videos. It's almost like physicians as a whole feel guilty for doing something else other than their life's work of taking care of patients. The thing I've stressed on this channel, I even spoken about it on a few videos recently, is that you have to take time for yourself and do other things outside of medicine. Explore other fun hobbies and whatnot. You have to take your mind off it. Yes, this is your life's work, but at the end of the day, you have to enjoy your life as well. It's very important, and it's how you not become burnt out. If in the next 13 years I develop arthritis or lose my hearing, do I regret not having taken the chance of getting my skills in place and I think the answer is yes. I think the answer is yes. That's what I hear. But see, I don't understand this because she's clearly set up going forward, especially for a retirement. Why is she asking if she should, well, I don't understand. Why doesn't she just do music on the side and also be a doctor? I mean, I have a full-time YouTube career and also working 80 plus hours a week in fellowship right now. In my job going forward, I'll have at least twice as much time, maybe like four times as much three time to do other things. So I don't understand why she can't just do music on the side. It's really odd. Something we're missing here. I wouldn't call it guilt or shame. But what would you call it? Lost opportunity. Yeah, all right. There's a high chance I would regret it. I mean, there are other outlets for music. It's just that our local orchestras and string quartets aren't meeting currently. All right, let me ask you this really quick. Let me ask you this. What kind of a doctor are you? General practice is a specific type of medicine you do? I'm an OBGYN, I deliver big. I almost said OBGYN, but that's a low salary. Oh, it's because it's California. For as high as it costs to live there, their salaries just do not match it. It's very similar to New York City actually. The salaries in New York City, most of them are academic. The salaries just do not match the cost of living here. You can go outside in like the Southeast or Midwest and make easily twice as much as you would in New York City, but New York City cost of living is crazy. And so is California and San Jose. I think you're at a position professionally and financially where you could draw back a little bit. I think you're practicing half. I agree with that. And pursue the music. And get the best of both worlds. She probably has patients that she's seen for decades if she's been in practice that long and you feel bad leaving that practice because then all of a sudden those patients have to do a find a new doctor and they've already connected with her and you feel bad making them leave and go do something. But at the same time, you have to take care of yourself. So I don't know why she doesn't just go part time. She clearly doesn't need the money because she's financially set.  Pursue music and do something you like.  You could quit your job and you know, you figure out a way to live on $3.7 million. I think you struggle through here. The only reason I wouldn't quit completely is I also think you'll regret that because I think you get a level of joy out of this. The interesting thing is she has $3.7 million in retirement. So there's a rule, the 4% rule. You can continue to withdraw 4% of your retirement account per year and it won't affect that principal balance. So she could essentially live off 4% of 3.7 million which is what? Let me see. Which is $148,000 a year roughly. She makes $200,000 a year now. She could pull 4% of her retirement every single year for the rest of her life at $148,000 a year and never have a problem. So I don't really understand this. It's almost like it's kind of braggadocious. I don't know, it's weird. This is a weird problem. I used to make over $300,000 which was my base pay and I have given up operating which is doing surgery on women and as of last year I did give up labor and delivery. See that's why her salary is low because that's what I was confused at. And then a lot of OBGYNs do this.  So she's cut down on surgery, she's cut down on delivery and it makes $200,000 a year. I don't understand what she probably has all the free time in the world. I'm really confused. So I'm no longer on call on weekend nights and holidays. So I went from full time which is in doctor's world 80 to 90 hours per week to start time which is about 50 to 50 hours per week. And I would parse it out in some way where you cut back to, I don't know, whatever you wanna call it, 30 hours or whatever. I would keep my toe in that water though. I don't think I would walk away cold. So obviously I didn't watch that video before but that's a little strange. She was almost like asking for permission to go even further part time or work less so that she could take care of her parents and enjoy hobbies. That was really strange. She was like asking for permission. So this is what I cannot stress enough for people who are in medical training and those who are becoming a physician. We worked so hard in our job throughout residency, throughout medical school, throughout fellowship.  Travel, pick up a new hobby, meet people, socialize, go out. Yes, we take care of patients and we should devote our life to that but it doesn't have to be every hour of our life and you shouldn't feel guilty for not doing that every hour of your life. So I just wanna stress that enjoy your life too because at the end of the day you have to take care of yourself before you can take care of other people. On that note, make sure you smash the like, subscribe button, follow me on Instagram and TikTok if you don't already and I'll see you all on the next video.


Live from Miami Beach, Florida, it's theCUBE, covering Veeamon 2019, brought to you by Veeam. Welcome back to the Fontainebleau Miami, everybody. My name is Dave Vellante. I'm here with my co-host for this segment, Justin Warren. Justin, it's great to see you. This is theCUBE, the leader in live tech coverage. Day two of our coverage of Veeamon 2019 here in Miami. Jason Buffington at Jbuff is here. He's the vice president of solution strategy. Congratulations on the promotion and great to see you again, my friend. Thank you very much. And Nathan Hughes, who was the IT director at Flexingate. Great to see you, thanks for coming on. We'd love to get the customer's perspective, so welcome. Great to be here. Okay, so Jason, let me start with you. Former analyst, you've been at Veeam now for long enough to, well, A, get promoted, but also get the Kool-Aid injection. You're wearing the green. What are the big trends that you're seeing in the market that are really driving this next era? What do you guys call it, act two of data protection? So, you know, I preached on this even before I joined Veeam, that every 10 years or so, when the industry shifts the platform of choice, the data protection vendors almost always reset, right? The people that led in network don't lead in Windows. The people that led in Windows didn't lead in Vert. The next wave is we're moving from servers to services, right, we're going from on-prem and to cloud. And so, and every time, the problem is the secret sauce doesn't line up, right? So, you got to reinvent yourself each time. And what we saw in the past generations, what we learned from is you can't be so busy taking care of your install base as you forget to keep innovating on what that next platform is. And so, for us, act two is all about cloud. We're going to take everything we know about reliability, but we're moving it into cloud. The difference is that in virtualization, there was one hero scenario, VMs, right? This time around, it's IaaS, it's SaaS, it's PaaS, it's using cloud storage, it's PaaS and DraaS. There's not a single hero scenario, which means we have a lot more innovation to do. That's round two. And you made that point today. You used the Archimedes quote, it could be a lever in a fulcrum, I'll change the world. You used the analogy of backup is now becoming much more than just backup. It's data protection, it's data management. We're going to get into that and test some of that with Nathan. So, Nathan, tell us about Flexingate. What does the company do and what's your role as IT director and tail? So, Flexingate is a tier one automotive supplier, which means that we provide parts, most of the things that go into a car, besides electronics and glass, to the final automotive makers. So, most of the companies that you're familiar with when you go to buy one. Okay, so you guys are global. I think you got, what, 24,000 associates worldwide, 64 locations. So, what are some of the things that are fundamental drivers of your business that are rippling through to your IT strategy? Well, our business has varied in the sense that we do a lot of different things in-house. So, we do obviously manufacturing, that's a big part of what we do. And then, even that is broken down into sort of different kinds. And then beyond manufacturing, we have advanced product development and engineering, and so we do a lot of light in-house. You support it all. So, you got diverse lines of business, you've got different roles and personas, engineers versus business people, versus finance people, and you got to make them all happy. We got to make them all happy. So, one of the things I love about manufacturing examples is, if you think about it, it's the two extremes of high-tech and low-tech. On the low-tech side of things, you've got this manufacturing floor, and it's just producing real stuff, not the zeros and ones that we live with, real things come off this line. And then you have the engineering and R&D side, where they are absolutely focused on stuff that comes out of some engineer's head into a computer, which is truly unique data. So, one of the things I love about the story is, talk about the downtime challenges you have around the manufacturing floor, because I learned some things when we first met that I think is phenomenal when it comes to manufacturing things that I didn't realize. Sure, so we have a lot of different kind of manufacturing environments. Some of them are more passive and some of them are more active. The most active environments are a form of manufacturing known as sequencing, and it's sort of where you bring final assembly of parts together right before they go to the customer. The way that customers order up parts these days, it's not like they used to back in the 70s and 80s where they would warehouse huge volumes of everything sort of on their site and then just draw it down as they needed it and you just kept the queue full. Now they want everything just in time delivery, so they basically want parts to come to the line right when they're needed and actually in the order they're needed. So, a final car maker, they're not necessarily making 300 of the same thing in a row, they're going to make one of this in blue and one of that in red, and they're all going to be sequenced behind each other one right after the other on the assembly line, and they want the parts from the suppliers to come in the exact right order for that environment. So, the challenge with that from our perspective is that we have trucking windows that are between 30, maybe 60 minutes on the high end, and if anything goes badly, you can put the customer down and now you're talking about stopping production at a Ford, Chrysler, GM, whatever, and that's a lot of money and a lot of other suppliers impacted. So, this is a data problem, isn't it? Yeah, it definitely is, and it's an interesting point because you talk about sequencing, Veeam has their own kind of sequence about how customers use the product and they start with backup, everything starts with backup, and then they move further to the right so that you get ideally to fully automated data protection. So, what are you actually using Veeam for today and where do you see yourself going with Veeam? So, right now we're using Veeam primarily as backup and recovery, that's how we started with it. We came from another product that was great conceptually, but in the real world it had terrible reliability and its performance was very poor as time went on and so when Veeam came on the scene, it was a breath of fresh air because we got to the place where we knew that what we had was dependable, it was reliable, we got to understand how the product worked and to improve the way that we'd implemented it. And so one of the key features in Veeam that really actually excited us, especially in those sequencing environments are these instant recovery options, right? So we were used to the idea of having to write down a Veeam out of snapshot storage and then being put in a position where it might take an hour, two hours, three hours before you could get that thing back online and now, or again to be able to launch that right out of snapshot storage was a blessing in the industry we're in. Yeah, did you see the tech demo yesterday where they were showing off how you could do an instant recovery directly from cloud storage? Yes, yeah. Does that get you excited? Yeah, no, no, that is exciting. Yeah, are you using cloud at the moment or is this something that you're looking to move towards? Cloud is something we're sort of investigating but it's not something that we're actively utilizing right now. Right.  I was talking to Danny Allen yesterday offline about it. He claims it's unique in the industry that, and I asked him a question I said specifically, if you lose the catalog, can I actually get the data back? And he said, yes. And I was like, that sounds like magic. So I guess my question, maybe both of you, is instant, how instant and how does it actually work? It just works, isn't that? It just works. It's just magic. Can you tagline? I guess we don't have to get into the weeds but when you say, when I hear instant recovery, we're talking instantaneous recovery with very short RTOs. To us what that means is that in practice, we can expect to have a VM from snapshot data back into production in about a five minute window. Five minutes. And that is sufficient for our needs in any environment. Okay, so now we're talking RTO, right? And then what about, so we said 64 sites across the world, 24,000 associates. Is Veeam your enterprise wide data protection strategy or are you kind of rolling it out now? Or where are you at? No, Veeam, I mean we started with a handful of key sites and we were using it specifically to back up SharePoint and a few other platforms. But once we understood what the product was capable of and we were sort of reaching the end of our rope with this former product, yeah we began an active roll out and we've now had Veeam in our facilities for five, six years. So you kind of swept the floor of that previous product. Yes, yep. And how complicated was it for you to move from the legacy product to Veeam? It was a challenge just rethinking the way that we do things. The previous product, one thing that I really had going for it, if this could be considered a positive I guess, is that it was very, very simple to set up. So you could take an entry level IT administrator and they just next, next, next, next, next. And it would kind of do all the things that they needed it to do. But the problem was that in the real world, that was sort of the Achilles heel because it meant that it wasn't very well customized and it meant also that the way that they developed that product, it became performance, it had poor performance. So the reason I ask that question is because so many times customers are stuck and it's like they don't want to move because it's a pain. But the longer they go, the more costly it is down the road. So I'm always looking to IT practitioners like, advice that you would give in terms of others, things that you might do differently if you had a Mulligan, I don't know, maybe you would have started sooner or maybe there was some things that you do differently. What would you would have done? Well, yeah, I mean, if we'd understood the whole context of what was happening with that other product, we would have moved sooner. And the one thing that I will say about Veeam is that it is, it's not click and point. It does involve a little more setup, but the Veeam team is excellent when it comes to support. So there's nothing to fear in that category because they stand behind their product and it's very easy to get qualified technicians to help you out. Is that by design? I don't know if it's by... I mean, well, the being great to work with, yes, that's by design. Yeah, yeah, but I mean... I was talking to Danny yesterday and asked about the interface thing because there is that always tension between making it really, really simple to use but then it doesn't have any knobs to change when you need to. That's why I'm asking. Yeah, it's complex either. Our gap actually comes a little bit later in the process, right? So you asked earlier about in what ways to use Veeam and we think of Veeam as a progression, right? So everybody, if they're using Veeam at all, or using it for Veeam backup and replication and because foundationally, until you can protect your stuff, right? Until you can reliably do that, all the other stuff that you'd like to do around data management is aspirational and unattainable at best, right? So we think the journey comes in that, yeah, it is pretty easy to go next, next, next finish. There's a few tweaks, right? To get backup going. But then when you go beyond that, now there's a whole range of other things you can do, right? I'm sure talked about data labs yesterday, the orchestration engine. Those are not next, next, next finish. But anything that's worthwhile takes a little bit of effort, right? So as we pivot from now that you've solved backup, then you can do those other things and that's where we really start going back into something which is really, really more expertise driven. As well. Well, it's early days too, and as you get more data and more experience, you can begin to automate things. Yeah, absolutely. So Justin was asking, Nathan, sort of where the direction is today, it's really backup. You've seen the sort of stages where you're talking about full automation. Is that something that is on the horizon? Is it sort of near-term, mid-term, long-term? I mean, coming to the conference, our experience with backup or Veeam is primarily backup and recovery operations. But I've seen a lot of things in the last few days that have sort of piqued my interest, particularly when it comes to the cloud integration, that's being actively baked into the product now. And some of the automated API stuff that's being built into the product. Any place where I can get to where we simplify our procedures for recovery, that's a plus. So I'm really excited about the idea of the virtual labs being able to actively test backup on a regular basis without human intervention and have reporting out of that. Those are things that I don't see in any other product that's out there. You know, there's another piece of innovation that we should think through. And so we've talked about the sequencing side, which is where we focus on RTO. How fast can you back in the running again? And when you and I talked earlier, the example that we kind of worked on was, think like a zipper, right? You've got the bumpers coming into a line of cars, and if either side slows down, everything breaks. And at the end, by the way, is the truck, right? Everything has to come at the same time, at the same rate. If there's downtime on either side of the source, you're done. But that's an RTO problem. The engineering side for high tech is an RPO problem, right? You have unique stuff coming out of somebody's brain into a PC, and it will never come out that way again. And so when we look at backup and replication, that should be the next pieces to go on. And then as you mentioned, data labs becomes really interesting in orchestration, so. Yeah. Well, speaking of human brains, and you kind of touched on it, Nathan, that you came here to learn some things and you've learned things from different sessions. So what is it about coming to beam on that is worth the time for IT practitioners like yourself? Yeah, I think it's all those things. I mean, we were talking about beam doing, you know, backup and recovery operations fairly straightforwardly, you know, in terms of getting in, but once you see some of this stuff here at a conference like this, you get a better sense of all the more elaborate aspects of the product.  This is a good place to really learn the depth and the level that you can go with it. And you're like most of your peers here, is that right, highly virtualized? Yes. Is that right? A lot of Microsoft apps. Yep. And it's a mid-sized, global organization actually kind of bumping up into a big. Sure. Yeah, cool. I asked about the data problem before. It sounds like, you know, the zipper's coming together. That's some funky math that you got to figure out to make sure everything's there. So, talk about the data angle. How important data is to your organization? We know how much data, and data's growing, and data's the new oil, all those bromides. But what about your organization specifically as it relates to a digital strategy? It's a buzzword that we hear a lot, but does it have meaning for you, and what does it mean? Yeah, data is vital in any organization now. I mean, we were referencing earlier how you've got sort of low-tech in manufacturing, or at least people think of it as lower-tech, and then high-tech in R&D, and how those things merge together in a single company. But the reality is that all of that is data-driven, right?  You're putting rivets into parts. You're probably taking pictures of that now with imagers when you're in manufacturing, and you do that so that if you get 300 bad ones, you can see exactly when that started and what happened at the machine level, right? So we're just constantly collecting massive volumes of new data, and being able to store that reliably is everything. Well, the reason I'm asking is, you guys have been around for a while, and you're a highly distributed organization, so in the old days, and even still today, you'd build a, you'd get a server, you'd, for an application, you'd harden that application, you'd secure that box in the application, running on it, and you'd lock the data inside, and what my question is is, can the backup approach, the data protection approach, the data management, or whatever we want to call it, can it help solve that data silo problem? Is that part of the strategy, or is it just too early for that? Yeah, I'm, sorry, I'm going to get you to repeat that question in a slightly different way. Am I correct that you've got data in silos from all the years and years and years of building up applications, and can you use something like Veeam to help unify that data model? All of them together, yeah. I think a lot of that has, it's more on the hosting side, right? So it depends on how those systems were rolled out originally and all that kind of thing, but yeah, as we've moved towards Veeam, we've necessarily rebuilt some of those systems in such a way that they are more aggregated, and that Veeam can pick them up in an integrated kind of way. You see that as a common theme, is this a Veeam as sort of one of the levers or the fulcrum to a new data architecture? We're getting there, so here's the trick. So first you got to solve for basic protection, right? But the next thing along the way to really get towards data management is you got to know what you got, right? You got to know what's actually in those zeros and ones. And so some of the things that you've already seen from us are around what we do around GDPR compliance, some of the things we do around sanitization of data for DevOps scenarios and reuse scenarios. All of that opens up a box of, okay, now that the data is curated, now that it's ingested into our system, what else can you do with it? When I talk to C-level execs, what I tell them is data protection, no matter who it comes from, including Veeam, is really expensive if the only thing you do is put that data in a box and wait for bad things to happen, right? Now the good news is bad things are going to happen, so you're going to get ROI, right? But better is don't just leave your data in a box, right? Do other stuff with that data, unlock the value of it, and some of that value comes in, now that I'm more aware of it, let's reduce some of the copies, let's reduce some of the compliance mandates, let's only put data that has sovereignty requirements where it goes, but to do all of that, you got to know what you got. Yeah. All right, go ahead, please. There was some impressive demo yesterday about exactly that, so we have the data, you can use the API to script it and you can do all kinds of it. Basically you're limited by your imagination, so it's going to be fascinating to see what customers do with it once they've put it in place, they've got their data protected, and then they start playing with things, come to a conference like this and learn, ooh, I might just give that a try on my data when I get back home. That's right. All right, we'll give the customer the last word, Nathan. Impressions of VeeamON 2019. Oh, it's been great, and like I say, you know, if you're a company that's been using Veeam even for a while, and you sort of have your entry level setup for backup and recovery, and I think there's a lot of probably companies out there that use Veeam in that kind of way, this is a great place to have a better understanding of all that's available to you in that product, and there's a lot more than just meets the eye. And it's fun, good food, fun people, that, thanks you guys for coming on, I really appreciate it. Yeah, thank you. All right, keep it right there, buddy. We'll be back with our next guest. You're watching theCUBE, Dave Vellante, Justin Warren, and Peter Burris is also here. VeeamON 2019, we'll be right back.


Hi there, I'm Emily from the learning team. In this daily SEO fix video, I'll show you how to use custom reports to monitor your site and keep an ongoing record of your site audits. Let's jump in. Within the custom reports section of MozPro, you have three options for creating your site audit report. The first would be to create it from scratch. Alternatively, if you have a medium level subscription or high, you can use one of the two templates. MozPro offers a full site audit report and a quick site audit report, depending on your needs. In this example, we're going to use the full site audit option. After selecting the template, I'll be asked to select whether I want to produce a monthly or weekly report followed by a request to enter in the title and description of my report. From here, I'll then be taken to the design report step setup where I can see that information regarding how new pages crawled, the total number of issues and a preview of those issues are included. This report type also includes information about linking domains I've gained or lost recently, as well as a breakdown of status codes for the pages crawled. This is a very thorough report that provides a great overview of the health of my website. However, I want to take this one step further. Moz recently added the performance metrics tool to Site Crawl, which provides information regarding how pages on my site are measuring up when it comes to core web vitals. And if you're not familiar with core web vitals, we have quite a few excellent blog posts available that talk about what they are and why they are important. With the addition of this tool, I can bulk analyze pages to see an overall performance metrics score for my pages on desktop and mobile. And now I can also add those to my site audit report. To do so, I'm going to head back to the add module step by clicking on the breadcrumb here at the top. I can then head to Site Crawl and then performance metrics. And from here, I can filter and select which modules I'd like to add in order to keep an eye on how my site is performing, as well as how changes I'm making to my site are impacting my core web vitals measurements. With the bulk analysis options and the custom report addition, this new feature makes it easier than ever to keep your team and key stakeholders up to date on your site's health and performance. I hope this was helpful. Thank you for watching.


Good morning. As discussed in the previous lecture, today in this lecture, we will start from this lesson, which is the first of a few lessons on the module of algebraic eigenvalue problem. I will again remind you that in order to follow the lectures in this segment, it is very important that the subject matter of this segment, this module is thoroughly immersed in your understanding. And therefore, it is very important that at this stage, you should have completed most of the exercises of this segment, because some of the background necessary for the following lectures is actually developed through the exercises in the book, in the text book that I have referred to you. In this tutorial plan, the problems of the book listed here, you must have completed by now, and that will help you in following the lectures in the coming module, which is chapters 8 to 14. Now, in this lecture, we will be studying eigenvalues and eigenvectors, in which we will talk about the eigenvalue problem as an introduction, and then generalized eigenvalue problem, which will also expose you to one of the practical problems from which eigenvalue problems emerge. Then, we will discuss some basic theoretical results, which will be utilized later for sophisticated methods of solving the eigenvalue problem, and then towards the end briefly, we will discuss a quick and easy method of solving the problem, which is power method. To begin with, I again draw your attention to the mapping A, which is from R n to itself, that is from n dimensional space to itself. That means, it is the corresponding matrix is n by n, it is a square matrix. Now, when we multiply a vector to a matrix to a vector, the vector gets mapped to another vector in the same space in this case, but then in this mapping, there are two effects produced on the vector. One is a magnification, which may be less than one, which means in that case, the actual vector will get reduced in size. The other than magnification, the other effect is turning rotation. Now, this is the general way in which a vector can get mapped through multiplication with a matrix. Now, some of the vectors for every matrix are special. They are special in the sense that they undergo only magnification or scaling, and do not rotate under multiplication with a particular matrix. These vectors are in some sense the own vectors of that matrix or some special vectors for that particular matrix, and these vectors are called Eigen vectors. The word Eigen in German means special or own. So, as if these vectors belong to this particular matrix. So, if you multiply the vector a matrix a to one such special vector, its own vector, then the result the mapping is nothing other than a pure scaling. So, in that case, we call this vector v as an Eigen vector, and the scale vector lambda is called the Eigen value or the characteristic value. Together, lambda and v Eigen value and Eigen vector are quite often referred to as the Eigen pair. They form a pair. Now, determination of all the lambdas and corresponding v is Eigen values and Eigen vectors for a given matrix is called the algebraic Eigen value problem. Now, how we can find the values lambda and the corresponding vectors v from only this much? The process the underlying concept is actually very simple. You can take this lambda v on this side though you cannot write it as a minus lambda into v, because a is a matrix and lambda is a scalar. So, but what you can do is that this v you can write as identity into v, and then take lambda i and a together in this manner.  Then you will have this system of linear equations. Now, you will note that this system of linear equations is n equations in this vector or n variables, and these equations are homogeneous equations that is the right and side is 0. Then you know that for a homogeneous system of equations for the existence of non-trivial or non-zero solution, the coefficient matrix must be singular. That is the coefficient matrix must have a null space, and v will be actually a member of the null space of this matrix lambda i minus a. So, for singularity of this matrix you must set its determinant equal to 0. Now, you find that we have reached the stage where from a large number of unknowns we have suddenly reduced to 1 unknown. In this particular equation you had 1 scalar unknown lambda and 1 vector unknown v, which was n plus 1 total number of unknowns. Now, the condition that the coefficient matrix is singular tells you the determinant of the coefficient matrix is 0. Now, you have got a single equation in a single unknown. In addition you know that this side is a polynomial in the unknown lambda polynomial of degree n. So, then the question boils down to finding the roots of that polynomial to begin with or to find the solution of this polynomial equation. And we know that it will have n roots including multiplicities. So, the polynomial on this side is called the characteristic polynomial of the matrix A. And therefore, the corresponding equation this equation is called the characteristic equation and its solutions are the Eigen values. So, characteristic equation or characteristic polynomial will give you n roots of this nth degree polynomial. These are the n Eigen values and for each of them you will try to find the corresponding Eigen vectors. That part should not be very difficult, because as you insert those Eigen values one by one for every Eigen value sitting here you will get a homogeneous system of equations in which the coefficient matrix is completely known. All that you need to do is to find the null space of that known matrix lambda i minus a, which we have studied earlier. Now, we have been just talking about the number of Eigen values. Total number of Eigen values from this will certainly be n, but that may be repeated. For example, suppose we have got a 3 by 3 matrix for which the Eigen values may turn out to be 2, 2 and 4 that is possible. So, here the Eigen value 2 is said to have an algebraic multiplicity of 2, because it is occurring twice in this polynomial. So, this polynomial will be lambda minus 2 whole square to appearing twice into lambda minus 4, 4 appearing only once. Now, we also talk of geometric multiplicity. That is when we take this Eigen value lambda and try to insert it here and try to find V. Now, in this particular example, if the Eigen values are 2, 2 and 4 then as you insert lambda equal to 2 here and you try to find the corresponding Eigen vector V, you would expect that there may be up to 2 such vectors. One Eigen vector belonging to lambda equal to 2 in the first instance and the second one belonging to lambda equal to 2 in the second instance. You may succeed in finding 2 such Eigen vectors or you may not that depends upon the particular matrix A. That means that if the algebraic multiplicity of a particular Eigen value is more than 1, then that may give you 1 Eigen vector corresponding to it or 2 Eigen vectors or 3 Eigen vectors up to the number which is the algebraic multiplicity. That means in a larger matrix, if suppose an Eigen value say this Eigen value lambda equal to 2 in a 7 by 7 matrix appears 5 times. So, the Eigen values are 2, 2, 2, 2, 2 something else and something further of this structure. In that case corresponding to 2 Eigen value 2, when you try to find out the Eigen vector, you may find only 1 Eigen vector, you might find 2 or 3 or up to 5, more than 5 you cannot get. That number corresponding to that particular Eigen value, how many Eigen vectors you could find out that number is called the geometric multiplicity. Now, note this one is algebraic, this one is geometric. Algebraic multiplicity is appearing from this polynomial. How many factors lambda minus a particular lambda is appearing in this polynomial? How many times that is appearing? That is coming from an algebraic source and that is why it is called algebraic multiplicity. On the other hand, the number of corresponding Eigen vectors will span a subspace in the space R n of the dimension which is equal to the number of linearly independent Eigen vectors that you can find corresponding to that Eigen value and this description of this subspace that you are talking about that is a geometric entity. That is why that number is called the geometric multiplicity of that Eigen value. Now, note that when we are talking about finding Eigen vectors, different Eigen vectors, then in that context linearly dependent Eigen vectors are not considered different. That means, if you find one vector as an Eigen vector, then it is obvious that twice of that will be certainly an Eigen vector. So, that is not counted as different from the first one. Similarly, if you have already found two Eigen vectors corresponding to a particular Eigen value, then a linear combination of these two will certainly be an Eigen vector with respect to or corresponding to that same Eigen value that is not considered anything different. So, that means, when we hunt for Eigen vectors, we look for linearly independent Eigen vectors. Now, when it happens that for a particular Eigen value, the algebraic multiplicity and geometric multiplicity have a mismatch between them. There is algebraic multiplicity is higher, geometric multiplicity is lower. In that case, we call that matrix as defective. In what sense it is defective, what is the defect and what to do in such a situation, that we will discuss in detail in the coming lectures. When it is so that algebraic multiplicity and geometric multiplicity for every Eigen value is same, in that case we can do certain interesting things very easily. We can diagonalize the matrix. That means, we can change the basis for representation of this mapping in such a way that the resulting matrix representation for the same mapping, the same linear transformation turns out to be diagonal. That means, the directions get completely decoupled. So, such matrices are called diagonalizable. To recognize a diagonalizable matrix, the direct straight forward thing is to check the algebraic and geometric multiplicity of every Eigen value. If they match all of them, then that matrix is diagonalizable. If a single Eigen value has a multiplicity mismatch between algebraic multiplicity and geometric multiplicity, then that is not diagonalizable. In that case, the Eigen vectors cannot be decoupled. The space cannot be decoupled in terms of individual Eigen values in the same way as diagonalizable matrices. So, actually the diagonalizability is that way not a property of a matrix as such. It certainly is a property of a matrix, but it is actually the property of a much more fundamental thing underlying the matrix that is the linear transformation. So, diagonalizability is actually the property of the linear transformation for which the matrix is just one representation. Now, considering these things apart, does this outline and try it tend to suggest that Eigen value problem solution method is complete? It may look so because finding the determinant of a matrix in terms of lambda is something which we can think of. Then, setting that equal to 0 and getting a polynomial equation is something which does not sound very dangerous. And then solving a polynomial equation also is something with which we are acquainted. After finding the lambda, putting that here and for every lambda finding the corresponding Eigen vectors, that also as a part problem is not very difficult problem. But does it mean that all the discussion in Eigen value problem gets completed here? Answer is no. The reason is that when the degree of the polynomial equation goes very high in that situation, solving this polynomial equation is actually not very easy. In fact, for solving a polynomial equation, one of the very popular, one of the very used methods says that try to solve the polynomial equation through the methods of Eigen value problem. So, therefore, for solving an Eigen value problem, the polynomial equation solving as a sub problem is not a very attractive proposition because as the degree of this polynomial goes high, it will be very difficult to computationally solve this problem. Therefore, people look for other ways of cracking this Eigen value problem directly without first making a recourse to this polynomial equation solving problem. And in that attempt, mathematicians have developed a method of interesting tools to handle matrices and express them in canonical formations and take a lot of advantage from these theoretical developments into several fields of applied mathematics. And these interesting developments will be studying in the coming lectures, including this one. So, in order to make the ground for that, I will need to develop some basic theoretical results first. Even before that, it will be a good idea to see a practical problem from which Eigen value problem appears. There are many such practical problems in almost all branches of science and engineering, but Eigen value problems certainly turn up. One such problem is the system of mechanical system with free vibration. For example, if you consider a one degree of freedom mass spring system for which the dynamic equation is just this, where m is the mass and k is the stiffness of the spring. And then, you try to write the assumed solution of this equation in the form because you know what kind of a solution this will have, this will have a sinusoidal solution. And so, you try to write it like this and then, you differentiate it twice and insert in this. So, you know that twice differentiation of this will produce minus omega square sign a is a constant. And from that very easily, you work out the natural frequency of vibration in which this mass spring system will undergo natural vibration. Now, when you try to formulate and solve the same problem for a multi degree of freedom system, we do not get such a nice simple scalar equation, but we get a matrix vector equation in this manner. So, free vibration of an n degree of freedom system will be governed by this equation, where m is the inertia matrix, k is the stiffness matrix, x is the vector representing the coordinate equation. And x double dot is certainly the acceleration corresponding to that. Now, in this problem, when you ask this question, what are the natural frequencies in which this particular mechanical system can execute natural vibration. And correspondingly, what are the vectors x along which those vibrations will take place. For example, in a 3 degree of freedom system, it might happen that x 1, x 2, x 3 give you a particular direction, a particular vector along which the vibration takes place in one frequency. There is another second direction in which the system may vibrate in a second frequency. Similarly, a third direction with a third frequency. So, what are these vibration modes and what are the corresponding frequencies? That becomes the problem for solution in this free vibration problem. Now, again in analogy with this equation, we now try to assume something vector x is equal to an amplitude vector into a term like this. So, there we assume a vibration mode first in this manner. The vibration mode x is a constant vector phi into sin omega t plus alpha. Again, we differentiate it twice with respect to time and insert that x double dot here. And that will tell us that this whole thing is equal to 0 because sin omega t plus alpha after twice differentiation will produce a factor of minus omega square. So, that minus omega square gets multiplied here. We have got this. Then, the same argument we use what we would use in this case that is for this to be equal to 0 for all time, this part has to be 0 because this one will not be 0 always. So, this has to be 0. When we do that, then we get the corresponding equation k phi equal to omega square m phi. Now, this resembles the identity problem that we discussed just now. In the earlier case, we got a problem of this manner k phi equal to lambda phi A x equal to lambda x or A v equal to lambda v. This is the kind of problem that we have been discussing just now. Now, here it is this problem is not exactly the same as this problem because in this location, there is a matrix sitting. Omega square you can identify with this lambda, but here there is a matrix sitting. That is why this problem is not called just identity problem, but it is called the generalized Eigen value problem as if in the original Eigen value problem, there was a matrix here which was identity which indeed we inserted when taking it on the other side. Now, in this case, in this particular case, it is generalized in the sense that in place of identity matrix, now there is a non trivial matrix sitting there. Now, how to solve this problem because if we take it on the other side, then I mean in place of i, if we have m sitting here, then as we take it on the other side, we will get k minus omega square m that will be the matrix, not the straight forward a minus lambda i as we would get in the ordinary Eigen value problem. Now, how to handle this? One might suggest that if we pre multiply both sides of this equation with m inverse, then immediately we get this problem m inverse k phi equal to omega square phi. Why not solve this problem? Because m inverse k we can take as a, we know m, we know k, we can evaluate m inverse k and then it becomes an ordinary Eigen value problem. Indeed it is possible to do that, but then it is not a good idea. Why doing this is not a good idea? The reason follows from the nature of these matrices that appear in these locations. This is not just some matrix and this is also not just some matrix. This is an inertia matrix and this is a stiffness matrix. Such matrices when appearing in practical problem have certain structure. A stiffness matrix is always symmetric and inertia matrix is always symmetric and positive definite. Now, if we evaluate this m inverse k that may lose the symmetry that was originally there in the original problem. Now, it is not a good idea to take a step in the solution of a problem which actually makes the original problem difficult. Later we will study in detail how solution of a symmetric matrix Eigen value problem is actually much simpler and much more straight forward compared to a general non symmetric matrix. Therefore, it would be a bad idea to take a step which will spoil the symmetry of the problem as originally given. Rather we should try to take a measure which will utilize this particular structure. So, what we do is that we take this symmetric positive definite matrix m and recall that for a symmetric positive definite matrix there exists a Cholesky decomposition L L transpose. So, if we conduct the Cholesky decomposition of this matrix m in this form L L transpose and then conduct a coordinate transformation. The original coordinates phi are now transformed to this phi tilde through this L transpose nu basis. In that case when we insert this here then see how this will look like we have a phi equal to omega square m phi. First of all in place of this m we will write L L transpose. The moment we do that we get this L transpose phi which we are going to define as phi tilde. So, L transpose phi we are defining as phi tilde. Now on this side also we would like to have phi tilde because we are applying that coordinate transformation. So, if phi tilde is L transpose phi then what is phi in terms of phi tilde that will be found through the pre multiplication of L transpose inverse. Now when we do that we get L transpose inverse phi tilde. Now we say that we can get rid of this L by pre multiplying both sides with L inverse. As we do that from here L inverse L gives us identity and we have got this. Now notice that the original generalized Eigen value problem like this has been transformed to this problem k tilde call this whole thing as k tilde. Then we have got the new problem as k tilde phi tilde is equal to omega square phi tilde. So, in the new coordinate system in which phi tilde is the vector we have got an ordinary Eigen value problem in which this matrix k tilde is actually symmetric. Because k was originally symmetric on this side we have multiplied it with L inverse and on this side we have multiplied it with the transpose of L inverse that will preserve the symmetry. You can just take that it is transpose is itself L inverse k L inverse transpose as you take the transpose of this whole thing you get the same thing back. So, the symmetry is preserved. Now note here that when we wrote L inverse transpose or L transpose inverse for this it is not clear whether we are talking about this or we are talking about this. Whether we are talking about the transpose of L inverse or whether we are talking about the inverse of L transpose that is not clear in this notation. Still this notation is valid because in these two cases the result will be same and therefore this L with minus t here actually means any of the two because these two are always going to be same. Now this is one practical problem from which you get an Eigen value problem. There are many other situations in all of science and engineering from which Eigen value problems suddenly appear. Now we will start with some of the basic theoretical results of the Eigen value problem over which we will build up later methods by which to solve the problem. Apart from that as a by product of this process the theoretical results will also provide us with tools to handle matrices in nice elegant and canonical ways which is useful in many areas of applied mathematics wherever matrices appear. Now first is the first important result that we should always keep in mind is that Eigen values of the transpose of a matrix are the same as those of the original matrix. This is very easy because we know that determinant of a transpose is the same as the determinant of an original matrix and the characteristic polynomial is found just by the expansion of a determinant. So these are obviously the same of course Eigen vectors need not be same in general they are different. Next important point that we should remember is the situation for a diagonal matrix and a block diagonal matrix. You know what is a diagonal matrix? So suppose we have got a 3 by 3 matrix in this manner. These are all 0 these are all 0 and this is a diagonal matrix and it is very clear that these diagonal entries are actually the Eigen values of this matrix and the corresponding Eigen vectors are the natural basis member. For example, if you multiply 1 0 0 with this then obviously you will get a 1 0 0 which can be written like this. So that shows that you have a v equal to lambda v. v is 1 0 0 that means a 1 is an Eigen value and this vector e 1 the first natural basis member is the corresponding Eigen vectors. Similarly, a 2 and a 3 will be the other Eigen values with corresponding basis members corresponding Eigen vectors as e 2 and e 3 the natural basis member. Now this is obvious. Now if you say that this is actually a much larger matrix this block this a 1 is replaced with a matrix a square matrix this a 2 scalar is replaced with a square matrix and similarly this a 3 then what you get is not a diagonal matrix because this square matrix may have of diagonal entries that will not be a diagonal matrix but what you call it is block diagonal matrix which will look like this in which this matrix a 1 is filled up quite a bit. Now when you talk of Eigen values of a block diagonal matrix then there is a very interesting situation that the the Eigen values of this large matrix is the Eigen values of a 1 and the Eigen values of a 2 and the Eigen values of a 3. So if this is r by r this is s by s this is t by t and everything else outside these blocks is 0 then the r Eigen values of this s Eigen values of this and t Eigen values of this separately obtained can be all put in a list and this r plus s plus t numbers will be the Eigen values of this large matrix. And the corresponding Eigen vectors they are also very easy to find they are just coordinate expansion for example if suppose this small matrix a 2 has an Eigen value lambda 2 with the corresponding Eigen vector as v 2 then just above v 2 you put as many zeros as as required to fit the size of this matrix and below that you put as many zeros as required to fit the size of this matrix. And then as you multiply this you find that this gives you lambda 2 into that same old vector that means that the a an Eigen value of a 2 is the Eigen value of a also and the corresponding Eigen vector of a can be found through a coordinate expansion over v 2 as many extra zeros are required above and below you can put that and then you get the big vector which is an Eigen vector of this matrix large matrix corresponding to that same Eigen value. For diagonal and block diagonal matrices the situation is very simple the matter gets a little complicated when you talk of triangular matrices a triangular matrices a triangular matrix will have non zero entries here. But still the diagonal entries are the Eigen values because below that everything else is zero. So, when you try to write the characteristic polynomial you write lambda i minus this. So, you will get lambda minus a 1 lambda minus a 2 lambda minus a 3 something something something here, but below you have got everything zero. So, when you try to expand this fellow determinant you get you expand from the first column then you get lambda minus a 1 into something plus all zeros then that something again gives you lambda minus a 2 into something plus all zeros and so on. So, for a triangular matrix you will find that obviously the characteristic polynomial will emerge as a product of these factors that means that you have got the characteristic polynomial already in factorized form that immediately gives you a 1 a 2 a 3 etcetera the diagonal members of the original matrix as the Eigen values. But Eigen vectors is a different question for that you have to do a lot of calculations to find Eigen vectors Eigen vectors are not so obviously visible here. So, when we handle triangular matrices we talk directly in terms of the Eigen values only not Eigen vectors Eigen vectors can be found with some further processing they are not so obviously visible. Now, when we take a block triangular matrix that is if these scalars are replaced with matrices and there are big blocks of zeros sitting below that and big blocks of other entries perhaps non-zero many of them will be non-zero are sitting here then we have a block triangular matrix which will look like this. This is a block triangular matrix with 4 blocks block a square block b not necessarily square block 0 which is also not necessarily square it will be the size of the transpose of b and then block c which has to be square then you say that the Eigen values of this is the same as the Eigen values of a and the Eigen values of c. Now, for this matrix this statement the Eigen values of this large matrix is the collection of Eigen values of the matrix a and the Eigen values of the matrix c can be easily seen in a similar way in which we saw just now the result related to the diagonal matrix. However, here the statement is made only for the Eigen values and not about the Eigen vectors. So, if the matrix a has an Eigen value lambda with an Eigen vector v that is this then we can apply the complete matrix h over a coordinate extension of v 0 and then we find that the product gives us this.  However, when we try to ascertain verify that the same holds for c also then we cannot immediately apply it on a coordinate extension because that will create the confusion with this b because in the product the way this 0 helped in this case it will not help in the other case. So, in this particular situation what we do is that we take mu as an Eigen value of c and then argue then it is also an Eigen value of c transpose and then c transpose w turns out to be mu w for that mu for some vector w. And then we apply not h, but h transpose on the appropriate coordinate extension of w in this manner and then we find at the end that we get mu into this vector 0 w. That means mu turns out to be an Eigen value of h transpose then that will mean that mu is an Eigen value of h as well. Now, apart from these results there are a few points which we need to keep in mind which will be very useful in many of the methods. One is that if we add a scalar times identity to a matrix then all the Eigen values get shifted by that scalar value and this is called the shift theorem. This is very easy to verify and so I am not going into that I am leaving it for you. Then the other important issue that we must keep in mind is actually applicable only for a symmetric matrix that is for a symmetric matrix a which mutually orthogonal Eigen vectors. A fact that we will verify in the next lecture for an Eigen value lambda j with corresponding Eigen vector as v j. We find that if we construct another matrix b in which from a we have subtracted this part then this resulting matrix b has exactly the same Eigen structure as a. Eigen structure means same Eigen values with the corresponding same Eigen vectors except that the Eigen value corresponding to that particular Eigen vector v j is no more lambda j but it is reduced to 0. That means the information worth of that Eigen value only has been removed from a. All the rest of the information of the Eigen structure remains as it is. Now, this is an important issue to which we will come back after studying the symmetric matrices in detail in the next lecture. Before that I will try to expose you right now to an important quick and easy method for solving the Eigen value problem and that is called power method. This helps you in finding the Eigen values of a matrix when you are not interested in finding all Eigen values of a large matrix but you are interested in finding only a few largest magnitude Eigen values or perhaps the largest magnitude and the lowest magnitude Eigen value. Now, this is very quick and easy method easy to understand easy to implement but note that it will work only for those matrices which have a full set of n Eigen vectors that is which are diagonalizable and for which there is a single Eigen value which has the largest magnitude. That means that the largest magnitude Eigen value has a magnitude which is larger than all the rest that is not two are at the top only one Eigen value is at the top. In that case power method gives you the largest magnitude Eigen value very easily. What we do for that is first to understand the way it operates you consider that if the matrix A possesses a full set of n Eigen vectors then these Eigen vectors will span the entire space R n and that means any other vector x that you can think of can be expressed as a linear combination of these vectors in this manner. Now, it is a different matter that given a vector x we can choose any vector x that will have a representation as a linear combination of the Eigen vectors with alpha 1 alpha 2 alpha 3 etcetera representing the corresponding coefficients. Now, even though we do not yet know those Eigen vectors and the corresponding coefficients but we know this much that any vector x that we can think of that we can have picked up will have some representation like this with alpha 1 alpha 2 etcetera and v 1 v 2 etcetera currently unknown to us. Now, if on both sides we multiply with A the matrix then what happens on this side x is a known vector which we have picked up. So, we multiply A x we can work out the result on this side we do not know what is happening exactly the numbers we do not know in detail but we know this much that A v 1 will be lambda 1 v 1 A v 2 will be lambda 2 v 2 and so on that means through a multiplication of A whatever was the representation here now in the coefficients we will get any other additional factor of lambda 1 lambda 2 lambda 3 etcetera. If we go on multiplying the vector the resulting vector with A once more once more once more then after p such multiplications on this side we will have A to the power p x which is known which is the result of multiplying A p times over x on this side we will have alpha 1 into lambda 1 to the power p v 1 plus alpha 2 into lambda 1 lambda 2 to the power p v 2 and so on. If we take that lambda 1 to the power p outside then this will remain inside. Now, under the assumption that the lambda 1 eigenvalue is the largest magnitude eigenvalue and the next one is a little below that what will happen is that as p goes too high many many times it has been multiplied then that will mean that in that case lambda 2 by lambda 1 lambda 3 by lambda 1 all being of magnitude less than 1 after raise to large power all of them will tend towards 0 when p is sufficiently large and that will mean that after many such multiplications we will have a vector sitting inside this which is in the same direction as v 1 and then after that process has stabilized after that direction has been stabilized one more application of that same multiplication with A will mean that on this side an eigenvector is being multiplied with A and that will give you lambda 1 into that vector and that gives you the vector in the direction and the lambda 1 as the scale between two successive values. So, as p tends to infinity this fellow tends to this lambda 1 to the power p as for 1 v 1 then you find that after the process has converge then you will find that the result a p x compared to the result in the previous iteration previous step are two vectors which are in the same direction that means the ratio between the first component and the ratio between the second component and the ratio between the third component will all be same and that ratio is lambda 1 that convergence all n ratios will be same in fact that is the test that convergence has taken place. So, this way you quickly get the largest eigenvalue largest magnitude eigenvalue note that it may be negative for that matter it does not matter. So, you will get the largest magnitude eigenvalue and the corresponding vector will be the eigenvector. Now, we will make two points here one is that other than the largest if you need the least magnitude eigenvalue also then how to do that for this purpose we can use the shift theorem. So, how to find the least magnitude eigenvalue what we can do is that after finding this largest magnitude eigenvalue we see it is sign this is a ratio which may have a sign. So, whether it is positive or negative that it has been found here. So, if for example suppose that lambda 1 turns out to be positive say the largest magnitude eigenvalue is 23 then what we can do from the original matrix we subtract 23 from all the diagonal entries that is application of the shift theorem. That is we subtract 23 i from the original matrix that will mean that all the eigenvalues have got shifted leftward by 23 that is whatever was 23 earlier that becomes 0 now whatever was 21 earlier that becomes 19 and so on. In that case the smallest magnitude smallest algebraically that turns out now as the largest magnitude eigenvalue largest magnitude then we can apply the same power method once more and then we will find that which is the largest magnitude eigenvalue and then as we shift the thing back 23 steps on the right side then we will get the appropriate correct eigenvalue for matrix A with the corresponding eigenvectors. So, this is one way to find the largest and least magnitude eigenvalues which has a lot of practical significance. Now one more possibility of an important question may be that for example if we are not interested in finding all eigenvalues, but we are interested in finding a top few the largest magnitude once lambda 1, lambda 2, lambda 3, lambda 4 etcetera some say 6 of them 6 top eigenvalues we want to find out and the corresponding eigenvectors. There also for example the matrix suppose is 100 by 100 we are not interested in all the 100 eigenvalues and their eigenvectors, but only top 6 or a few top ones with some conditional requirements. Then what we can do after finding the largest one we can use deflation. This will work in the case of symmetric matrix which is quite often encountered in practical situations. By deflation what we can do is that we can subtract the part which is contributed by this particular eigenvalue lambda 1 and the corresponding eigenvector. Then the resulting matrix will have the largest magnitude eigenvalue as lambda 2 which can be found through power method and so on. Now this is a very straight forward method which can be applied if you are sure that the matrix does satisfy these requirements. Otherwise the process may not operate as expected or as desired. Now apart from these things there are two important concepts which will go long way in our discussion in the coming lecture. One is the eigen space. This is a term to in use for representing a subspace of R n which is composed by the eigenvectors of a matrix corresponding to the same eigenvalue lambda. For example suppose A has an eigenvalue lambda corresponding to which there are k eigenvectors v 1, v 2, v 3 up to v k. Then that will mean that any linear combination of these eigenvectors is also going to be an eigenvector. You can verify that very easily. Suppose corresponding to eigenvalue lambda there are two eigenvectors v 1 and v 2.  Then if we apply A on a linear combination of these two eigenvectors, then we will find that this will turn out to be A 1 is scalar.  And taking lambda scalar outside this common we find that we have got this. That means the matrix A multiplied over this vector gives us lambda into this vector. That means if v 1 and v 2 are two eigenvectors corresponding to the same eigenvalue lambda, note that it is applicable for same eigenvalue. Then any linear combination of them is obviously going to be an eigenvector with respect to further particular matrix A. Now, this is not a linearly independent eigenvector, but this is certainly an eigenvector. It does not come in the counting of eigenvectors, but whenever required this vector does operate like an eigenvector. And that means that if these k eigenvectors are corresponding to the same eigenvalue lambda then the complete subspace spanned by these vectors gives you a subspace in which every vector is an eigenvector. And therefore, this particular subspace is also called the eigenspace of A corresponding to that eigenvalue. There is another important theoretical point that will be quite in our discussion in coming lectures that is similarity transformation. This is something which we have earlier seen once and here we look at some important properties of it. If we decide to represent the vectors of a space R n in a different new basis S and therefore, the matrix representation of a linear transformation changes from A it becomes B, where B is S inverse A S. This we have seen earlier. Now, note that determination of lambda i minus A which is the characteristic polynomial of the matrix A. Now, we already know the determinant of a matrix and the determinant of it inverse are reciprocals of each other. That means that if we multiply this with determinant of S and also with determinant of S inverse we are actually making no change because this will be reciprocal of this. We also know that determinant of the product of three matrices of the same size is same as determinant of P into determinant of Q into determinant of R. Now, what we have got here is determinant of P into determinant of Q into determinant of R. That means this is same as determinant of P Q R. That means a single determinant with S inverse inserted from this side and S inserted in this side will be the same as this. Now, when S inverse and S are inserted on from the two sides on this they cancel each other because identity is sitting inside. That is why this is lambda i. On this it will have the effect which is different that is S inverse A S which is B. That means we have got this whole thing same as determinant of lambda i minus B. What is that? That is the characteristic polynomial of the matrix B. Now, that shows us that through the similarity transformation the matrix might have been changed but its characteristic polynomial remains same as earlier. The characteristic polynomial of A and the characteristic polynomial of B turn out to be the same. If the entire polynomial is same for A and B then all the roots will be same. That means that Eigen values remain unchanged through a similarity transformation because similarity transformation comes out only as a result of a change of basis. No geometrical entity is being changed only its representation is being changed and Eigen values are the property of the underlying linear transformation not of the basis. And therefore, Eigen values remain constant through all these similarity transformations. How do Eigen vectors change? Geometrically even Eigen vectors do not change but then their representation in the new basis will change as the vectors as any other vector would change its representation in the new basis through the multiplication of S inverse which we have already studied in the same manner and Eigen vector of A will transform to S inverse V in the new basis which is given by A. So, if V is an Eigen vector of A the corresponding Eigen vector of B will be S inverse V because there the new basis S has appeared. So, the basis change of vectors takes place through this relationship and the same will apply to Eigen vectors as well. Now let us quickly summarize what are the points that we have discussed in this particular lecture. First important point is that meaning and context of the algebraic Eigen value problem that we have discussed. Second is that we have studied the fundamental relationships deductions which are vital for the solution of the algebraic Eigen value problem. And third we have been exposed to a quick and easy method called power method as an inexpensive procedure to determine the extremal magnitude Eigen values only the largest or largest and lowest or the largest few. In all these situations we can use the power method with a little bit of help from the shift theorem or the deflation technique. But then while applying power method you must be careful that the power method does not apply to arbitrary matrices but on certain matrices having particular kinds of Eigen structures. If your matrix falls in that category then power method will be very handy for you in many situations but otherwise it may not operate as desired. So, in the next lecture we will build up on what we have developed till now and see the detailed discussion on the theoretical development on Eigen value problem which will be then used in different categories of methods for solving the algebraic Eigen value problem. Thank you.


Everybody forced here we've been 2023 top scrim update breakers to light 10 box case over 20 picker team style Anton with the Astros been aged Guardians Chris v. Orioles Chris S. Metz Stu Chris S is Chris Pretty close your Chris S from Michigan with the Mets Chris for us California with the Dodgers Kurt S. Braves Darren C pirates Dave B. Rockies A's and Giants Dell has the Royals and the Padres Ed B with the Phillies Eric E. Cubs Ian has the D backs in the Angels God James J. Marlins Jerry F. Cardinals Ken H. Yankees Matt S. Mariners Raj has the Blue Jays Ray G with the White Sox Richard V. Red Sox Rory D. Twins Ryan B. Ray's Stefan with the Tigers Tim W. Brewer's Victor F. Has the Rangers Wong S. Reds and William S. with the Nationals Let's go ahead and get started Which does work does one work better. Yeah, I did I use the silver Oh, she's over as he goes with it Does it so it works as a dry We don't know It's gonna be off by tomorrow don't get it too humid That's way more professional Doesn't even work super well. Hope it stays You got they're gonna walk out be like Not that with that. Yeah debut patch right here The JG it's not it's literally like I said a two-year-old could probably write that No, I've signed some cars like the national and stuff just random cars people bring up to me. This is way better Yeah, hopefully it dries and it stays What did you say he's break? Oh, yeah, this is a yeah. Yeah. Okay. Gotcha. Gotcha. All right. Here we go y'all Mom or fractures of Jared Schuster Miller debut Jeter downs BB debut Brenton Doyle Entity gold all-star Adola scarcia for Texas number to 50 2450 Purple Rook into 50 is Valdez for the Red Sox Aqua blue to 175 Solano twins a youth quake of Brito Blue auto blue rookie autographs act netto for the Angels Solid one there 102 of 150 for the Angels and rookie autograph eggy Rosario for the Padres There's Oda Rizzi and Bailey rookie refractor Bailey for the Giants those whites for the Angels Burleson debut Cardinals Valdez Red Sox Matt Moore Aqua debut Dodgers is Miguel Vargas to 199 Florida negative Dominic Smith Nationals Purple Wendell to 250 Marlins youth quake of Schuster for Atlanta Orange auto is Roger Clemens Clemens Stroh was auto 25 at 25 That's really cool. Yeah, they they're they're cheap in this too There you go, Tony. 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There's like a scratch next to his hip there We'll get a box ever for the O's Try to get down to replace blue refractor Rosario Braves the 150 got magenta debut of 399 arrays Taj Bradley Tampa Taj there we go Jokes youth quake Astros rookie auto massy for the Royals and a red and a red of Oswaldo Cabrera three of five Damn Yankees can there we go Ken true red on card and all Me or else we'll send a box and wrapper for you So the second museum the half case is at 19 to go First one is sold out. Thank you all very much for that Lot of football left black footballs all on sale all teams marked out finest flashbacks is on sale That's coming up pretty soon That's big time discounted and then only six to go on definitive now as well Yes, I just don't know when they will have a personal Zalmanek next time Fractor debut of Miller for the A's Perkins for the Brewers Phillips got Jake Alu Nats rookie refractor Tiosca Hernandez and Randy Vasquez rookie refractor for the Yanks Got a purple Adley debut 250 for the O's aqua blue Bush debut to 175 Dodgers Radiating rookies morale for the Cubs short print cubbies Go Eric E with the Cubs Good case here. 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Bladay for the A's A's 10 at 25 Magenta 399 Benintendi White Sox Youth quake of Michael Bush for the Dodgers Gold auto gold rookie autos Cal Stevenson for the Phillies number 250 other nice color in the in the break here Ed B with the Phillies and rookie auto Taj Bradley Tampa. Let's go guys a couple more Fracture of Ozarkar got a delay Chapman Bellinger wisely and a Rosario Orange red rookie lava lamp is Brett wisely number to 10 for the Giants Five or 10 Giants. That's like a color match to with them There we go. That's pretty sweet. Magenta purple Dustin made a 299 Dodgers True blue fulmer for the Cubs youth quake fought for the D.Bax a Purple rookie autograph brand and fought D.Bax 250 Pitcher chase there for the D.Bax E and J with them and rookie refractor auto A's Mason Miller 182 a 490 on a little behind we're gonna get into the next museum right after this the one after that still has space to go He's got to get it filled up Chrome update hobby rips before this bleached. That was a break before this fractures Martinez Volpe debut Yankees May Gonzales debut Guardians Solano and a Vargas debut for the Dodgers Magenta purple rookie to 299 Michael Bush Dodgers Aqua rookie to 199 Randy Vasquez Yankees Genta rookie net so to 399 angels Rockio youth quake guardians Purple rookie auto Bryce Miller for the Mariners here to 250 Seattle again Matt S with Seattle and rookie base auto of Ruiz for the A's Give me a second. We'll do a quick recap Steven with a Sunday fun days, but it's not a 20 now week 13 again 25 boxes 2019 NT and immaculate 2022 NT photo 2022 I'm act photo as well Appreciate it Victor. We're down to 18. Let's go. All right. So we had the great any rookies of morale gold of Adola scarcia gold fairchilds Got orange for the day for the A's the 25 To 10 we had wisely for the Giants orange and red lava lamp autos of Caballero Ruiz Rosario Tovar Massey Bradley Lee crook To 499 we had Miller Vavra Gavin Stone to 299 niner Diaz to 250 Miller and font to 199 Swarmer 150 Rockio and netto Golda 50 Stevenson a red rookie auto of Cabrera for the Yanks And then an orange to 25 was Clemens for the Shrews on the auto All right, y'all that'll do it. We'll get started on museum just a moment here