We're really glad to have you, thank you very much to Ricardo for introducing the work we have done thus far in the public information session. We have a bunch of folks, instead of introducing ourselves because we have so many, if you speak, please just introduce yourself and maybe something distinctive about you. We do have 20 minutes. This is our chance to listen to you, so my job as facilitator is to just make sure that we have everybody share the air and nobody goes on too long and we listen carefully. We have a scribe, so Peter is going to take down what you are saying, and if we have a chance at the end, maybe we'll just read through to make sure we've captured things faithfully. I don't know if you've ever been in a circle before or had a talking object, but we have two talking objects here, and you have to choose. This one is called C Sparkle, and this one is Salmonella, so I want everybody to say after me, we want Salmonella, perfect, we'll use this one. So the prompts for you all tonight are, what opportunities in this area haven't you heard yet, mentioned yet, right? So we have a sort of funnel of ideas that are coming into the commission, or things that we're aware of, but we are very cognizant that we don't know everything, we're not aware of all the opportunities out there, you are some of our best sources of information, so please think in that mindset. Are there other ideas you'd like to share, and then what would you like to be part of in this area, in terms of getting involved in the action, volunteering, stepping out or things like that? So, and then the last one to be thinking about is what do you think other people in the community need in this area? So I've just loaded you with all four potential questions, but I'm doing that now because we have 20 minutes, and we may not get to go through them in order. So, speak now, if I hold your piece, you have Salmonella. Pat, inkly. I haven't heard anything about hardening off buildings downtown, and it feels like that's really important to do an assessment of, you know, the M&T Bank did it, not totally successfully, but... Does that mean closing off from water? What is hardening? They put in structures to close off for the doors, and you can do it with windows too. I have Salmonella for a second. Pat, thank you for that. One thing that we are not going to do very much of is respond, because we really want to hear from you. Please trust that we are taking this in. And in some cases, if you see my jaw drop, you'll know we hadn't thought about that at all. If you don't see my jaw drop, you know, we made you aware of it. Did your jaw drop? It did not. Carolyn Hrudinsky, and I'm supposed to say something about myself? No. I think there's a lot of parking lots that could be made a little more resilient. Yes. Ken Jones, I have Salmonella. I want to build on that. It's the state office complex. It has changed tremendously since COVID. And the pressures that used to be those parking lots have been reduced. But I'm not so sure what the state is doing with regards to the vast real-estate holdings that they have. And can those be reconsidered, repurposed, rebuilt in such a way to allow the downtown to grow? John Copan, this is a very small idea in one of the subcategories, but I'm on the Wrightsville Beach Board, and people come from there all the time. That's a great space to do some education about the waterway and how we use the waterway. So thinking about interpretive materials at Wrightsville and other places where we access the water, not just in Montpelier. I just wonder about, you know, when we were talking about Montpelier used to be so thriving like at lunchtime with all the workers in the state office buildings and stuff. And I'm just wondering about bringing people back to those office buildings so that they're in that Montpelier and the merchants benefit from that.  But I haven't heard much about reopening the offices and, like, you know, what's going to happen to all those workers. So I'm really interested in what would happen with that. Your name is Tim? Oh, Tim. Yeah, Glenn. What's all you thinking? Sure. I'm Jack McCullough. I'm the mayor. And I... The questions that people are raising about the state workers downtown, we don't know either. And the state doesn't know. And we would love to see those workers back downtown spending money. But if they're not going to be downtown spending our money, we really want to see all their real estate put to something more productive than waiting for workers to park their cars who aren't going to show up. Well, I work in a building that has no elevator, hasn't had an elevator, and maybe never will. I don't know what's going on. People live in the building. It's not the only one. I'm sure everyone here knows about that. I haven't heard much about, like, how to build buildings without having massive infrastructure get destroyed like that. I understand, you know, putting in vinyl floors and moving stuff out of the basin, but an elevator is really important for a lot of people. And I haven't heard much about what's going on and why they're so hard to fix. What's your name? Oh, sorry. I mean, Sean. I mean, is it a building that had an elevator that is now destroyed by flood? Yeah. Yeah, it's both residential and office. Got it. Kara Robacheck. I'm reminded of the fact that my son and eight or nine of his friends boated to popular high school at the end of last school year. And it occurs to me that the river and this building are very connected, and I'm hoping that some of the education can go through the schools and can involve the students. Bear with me. In the Czech Republic, a mandatory school subject is rafting. River rafting on rubber boats. Because it was targeted as an area of cultivation for outdoor recreation. So I think very much in keeping with your comment, it's something to look at in terms of really developing the river as an attraction through like cultivated sports and activities and high school subjects involving inflatable rafts. I'm sorry. I can't catch this. I can't get rid of this. Started with state. I also want to mention the federal office building, which is the former home of the post office, will be renovated in some fashion. And so that's a prime opportunity to rebuild that in such a way that is flood resilient and meets the needs of the city in regards to housing, whatever we need in terms of other services and how, and I want to get to that state thing, Montpelier has a lot of loud voices. If we can use our loud voices to convince our state partners that they need to listen to us because we can be a pain, do that and then use that practice to them work on the fence. So there are a lot of parking lots in Montpelier. Perhaps some people think perhaps there are too many. They could be made into these nice bowls that receive a lot of water and they could be skate parks and playgrounds and all kinds of stuff when it's not flooding. Rain gardens. I'm just going to remind, one of our last questions is what do you think other people in the community need in the area of adaptive downtown? Sometimes it helps me to sort of have my own issues, but it helps me to think about trying to guess at what other people might need. Thank you. I'm Gary Holloway at Salmonella. One of the things that was brought up in the previous forum was just the need to kind of get out of this combined sewer stormwater systems that are really impacting the neighborhoods around our downtown during flooding events. So I just kind of want to bring that point up again. It really does relate to Salmonella. Yeah. Get this anywhere. Do you have E. Coli over there? Do you have E. Coli over there? I don't have that. It's tempting to bring a whole collection, isn't it? So who let me ask a different question. How many of you have traveled even, say, to Barrie or much further afield into places that are connected to their rivers and have probably experienced or may experience what we just faced in July? And did you observe anything about those places that we could use? I see Pat. Oh, darn it. You flinched. You brought the painting. Venice. And Venice has a lot of water around it. And Venice has done a lot of what's called wet proofing to prepare for the water and just understand it's coming, it's not going away. I couldn't tell you what that was, but they've adapted to the fact that the water is rising and it's not going to change. Have Salmonella. I mean, Sean, I was in Oh, great. Let's hear the personnel line just since we're back to it. Hang on just a second. Hi, Elizabeth Parker here. I'm curious about what is being done to collect information on hydrological changes that have happened in downtown. For instance, the post office supposedly has a stream that is surfaced and is across the street from Clay's Church. You know, we're curious about what is happening with that and have not been able to get any information from the people who supervise that building. So we've also experienced a lot of structural damage of sinkholes that happened that caused natural pipes to bore through the foundation. And I'm just curious what information is being gathered on that nature around town so that we can understand and share with one another how the formation of the water underneath our buildings is changing because it's going to have a big effect on us as we go forward. Thank you. Thank you. This is Nathan speaking. Great question. So one of the things that I uncharacteristically responding, one of the things we're aware of is that the state is about to embark on a scoping study for this area about the Wundewski River and things like that, and we can ask for sure whether that will include the kind of question you're asking. So thank you for raising that. Yeah. I took it from you. I don't know why I took that from you. No, that's okay. You wanted to talk. I just wanted to get back to your prompt earlier. I was down to Jersey Shore last summer, two weeks after the flood here. And in a place that is a very thin island and gets flooded quite often, and they have made major requirements of any new construction that everything has to be well off the ground. And so what you see is houses that are built on pylons and parking underneath. Nobody has a ground floor entrance that's been built in the last 15 years, I don't think. I don't know what this town would look like, but it works and it's interesting. So it's something to think about. Before I pass it back to Jack, we are close on time. I'm sure we're going to hear a bell soon. You don't have to leave. If you don't wish to go to another group, you can hang in and see what the next group says. We'll see what we have for turnout. But if you wish to, at the bell, Watershed is back in the auditorium, and response is right next door over here. Jack. Thank you. Jack, look out again. I just want to point out that there are things that we can do right. There's a lot of damage downtown, but the new buildings that were built after our new code was put in place got through it. They didn't flood city center building, which was built back in the 1980s. And the transport transit station and apartments and there are a couple of other places like that that didn't flood. So we can be adaptive and plan for survivability. That's part of what we really need to keep doing. Thank you. There's a small city in Maryland that gets significant floods every five years, and it's a very historic town. It's not an atlas. It's in a river. What is it? Ellicott or Indicott? Yeah, Ellicott City. They've been able to maintain their historic buildings, regularly. So I don't know what they're doing and it's a tragedy, it's horrible, but they're still there. All right, so my last prompt if we have any time, and you can also respond to this in writing on the posters out there, email, any way you like. One of the things we are very conscious of is that the only power we have is given by you. You sort of called this into being. A year from now, two years from now, five years from now, if we have had an impact, we as a commission and we as a commission working on your behalf. We're curious to hear if folks can sort of visualize forward. Well, I wrote a whole story and it's out there if you want to read it about what happened and what comes next, what came in the future when we did these things. It's on the table, there are a few left. I gave them to all of you all in the council. Thank you so much. Paul Burns. I think it won't be that there are no more climate catastrophes. We don't have that power here, but that the actions that are being discussed here and in the different groups are having an impact, so therefore the flooding is less than it would have been in our downtown and buildings and other structures are not as vulnerable. Thank you so much. Even if you're standing, just get up, it's good for you. Yes, good to see you. You can just write it down on the side. Folks who care about Monculier do not have to sit, but you are welcome to find a seat. My name is Nathan Souter. I'm a member of the commission. I'm grateful that you are here tonight. We are grateful that you are here tonight. I'm going to be the facilitator. We have a scribe, so Peter Walk is writing down everything that you say. He's probably embellishing a whole bunch. Ricardo, spoke to you from the stage. Ben Doyle. We have Katie Trouts. We have Greg Gossens. Paul Carnahan. Okay, so we want to hear from you. We want your ideas. We want your understanding of new opportunities. We want anything you can share with us. I'll read you a couple of prompts in a minute. We're not going to be doing much responding. So this is really intake. Peter's going to record it. We are aware. We have sort of an idea funnel or a project funnel. So we have things that we're aware of what we should do about. But we're starkly aware that we don't know everything. And in fact, people who live here in this city who have numerous connections and networks are great resources. And that's you. So please give us what you got. To the prompts, I'm going to read all the prompts that I want you to respond to.  Respond to anyone they feel good about. What are opportunities in this area that you haven't heard about yet that's the adaptive downtown area? Are there other ideas you'd like to share? What would you like to be part of in this area? So if you're going to take action, volunteer, step up, how might you do that? And then what do you think other people in the community might need in this area? We are going to, if you've been in a talking circle or use a talking object we have two to choose from. This one is called C-Sparkle and this one is called Salmonella. So I want everybody to say after me, we want Salmonella. Perfect. We'll use this one. Alright. So if I pass this to you, please introduce yourself and then let us share your information. I will do a little bit of managing if folks are going on longer. We want to hold space for everybody to move. Anybody can catch Salmonella. Thanks. My question is it's wonderful to plan ahead and I love all the information you're gathering and posting because we all need to learn. But I'm very anxious about not having any focus on some immediate resources. I mean, I read a lot about some places that actually put out these big, huge inflatable they put water in these tubes to have immediate way to keep water out of their stores. I'd like to see some fundraising and projects that deal with if it happens tomorrow I think pull out of the garage and put in front stores, doorways. Thank you. Thank you. I'm curious. Who are you? I'm Dan Jones. I am curious on this adapting existing buildings to flood resilience. Has there been a lot of research on that because it sounds like something that if we have it two or three times, it kind of the underpinnings it kind of gets into the wrought area and so I guess it comes down to the idea even though we have a beloved historic downtown is it really possible to insulate it? Yeah, actually sorry, Barbara Connery in response to your question Dan, yes there are quite a few resources from FEMA about things called wet flood proofing dry flood proofing we can protect our buildings you know, many of you are aware of the former Chittenden bank building now the M&T bank that has a giant flood gate at their entrance those are the kinds of things that I think we need to be looking at and producing we need to keep everything out of basements we need to keep mechanical systems out of basements so I think what I'd be interested in seeing is that we develop a checklist or some kind of working plan for the existing downtown and it may take looking at each building individually by developing a plan for how we can address flood resilience for each of the buildings and dealing with that I kind of agree but I would Sal, I'll follow I wouldn't want to put all our eggs in one basket and I've I've been looking at redevelopment of the downtown in what I call the pit I guess people notice the pit but you know the giant parking lot a big donut hole in the middle of Montpelier which is a complex real estate situation the time might be right to approach the competing ownership of that area so that we have a backup and as the city expands I mean we can expand into that historical district downtown it seems to me the ideal place to expand downtown is into that centralized could you just find the area you're talking about because the pit first starts from different places to the east of the state house the parking lot behind the thrush the post office Court Street this is what I love about Montpelier, he just said behind the thrush so the pit that Sal is referring to is bounded by Court Street the street that runs up past a pavilion State Street East State Street so that's sort of invisible area back there thank you alright I'm Rob Goodwin I think on that front talking about new construction downtown I think one interesting point here is that the biggest development downtown that didn't get built years ago was the new hotel and parking garage if you look at the water elevations of where things came and how that would have fared and how that was designed it would have been fine just doubling down on the fact that new development is really our opportunity to create the space and to create the economic activity and the housing resources in this town that are safe and also it's like a project like that we don't have a hotel right now there's no place for people to come to town go patron these restaurants at which lost a lot and need the business to really get back on their feet and so that project didn't happen I think for some complicated reasons everyone know here but I think we should remember that new development in this town although may not be the exact thing that we want it is an opportunity to not be in the flood plain even if it is built in the flood plain we can build it right from the beginning Paul told him that I wouldn't catch it so I'm Nancy Boone and I just wanted to throw out something in relation to what's out there that can help us know what the possibilities are the National Park Service has done an amazing amount of work with historic areas around the country and looking at how people are preparing for a flood how they're responding what kinds of physical changes are being made and they even though they're the National Park Service they're happy to assist communities and I've worked with several people at the Park Service who I'm sure would love to come and give a public program so that could be fun and just sharing information and building on our optimism so nice to see you can I go again? hold on a second is there anybody someone on the so I'm gonna stand against the wall there I can pause for a zoom question if there are some okay my name is Alaya Mourning and I grew up here in Montpelier I went to this high school and I recently returned to raise my young family here and one of the things that I'm really thinking about as we have these conversations is we love our historic downtown there's like a nostalgia inherent in this place and it's really clear that we're gonna have to make some really dramatic changes if we want to keep living here like the fact is we cannot keep doing it the way we've been doing it so I I'm curious about what are the conversations we need to be having within the community to get people ready for that reality amen I'm still Sal Fano everybody here seems to be interested in this topic obviously and after after the flood I did a little research on search terms like watershed management and that sort of thing and I discovered a report called the Vermont economic resilient initiative which was published in 2015 by a group like this one called Ireen which happened in 2011 so four years later and in it I mean it might be worth a read I think it's available on the PDF form or online because there are there's a focus on five projects where they they adapted existing infrastructure or they changed the their relationship with the river so several different solutions to problems similar to what we have and it gives you a good idea of what can be done and what it looks like and how it works so again Vermont economic resiliency initiative DERI I've seen that report that isn't good just to piggyback further some of the images you saw in the slides of architectural drawings or renderings of downtown or various river corridors those came from other movements like that that let's think about the capital area let's think about this river so there are a lot of concepts there are a lot of projects that have been at least studied enough to create drawings and action items and then they're waiting for those and we have Paul Cronahan knows a ton about the history of this area Greg Gossens has been involved in some of these projects so we've got some of that but there could be things we're missing so thank you for that that's great in reference to that Nathan the Sustainable Montpelier Coalition did a pretty extensive competition to redevelop the downtown sustainable and also more energy efficient way and so many of those addressed the whole issue because we didn't let them off the hook about our water issues and so many of those addressed expanding the water the floodplain and allowing for that to happen the big challenge was however and I'll bring this up is yes new buildings are great and they are above the floodplain but if you start talking about the big parking lot behind the state buildings the flood level is at least 10 or 12 feet above that that level so if we start building buildings that are elevated 10 or 12 feet above grade then what does our downtown look like Kasia you may have missed the introduction but anyone can catch someone else I picked up on that Hi my name is Kasia I am curious to hear what the state is doing and how the state intends to lead the way on their land that they own the state owns a significant portion of downtown and from what I see I do not see the state following their own best management practices when it comes to how they treat the river how they build buildings flood resilience and I would love to see the state demonstrate to us through their actions exactly and would love to see our community encourage the state to do so encourage is a nice word I would say fresh we are not responding but just to quickly say that is so key 40% of downtown is pavement controlled by the state and just so you know Pat Moulton I don't know if you know her but she is here she is the central Vermont recovery officer and Pat's had every great job in Vermont when it comes to community and economic development and get stuff done and we have definitely been a part of conversations already between city, BGS, Vermont Mutual there is obviously an opportunity with the federal building that there is just a lot of chess pieces and I think the state absolutely can demonstrate leadership and what a truly resilient downtown can look like and unlike Barrie where you have 40 individual homeowners and you have some grand vision the state could just negotiate with themselves and make that happen maybe maybe I just want to on Cassius word encourage and hold focus on their own standards one of the things that we could use your thoughts and support on you know each of these different partners we call them partners maybe a different approach different connections, different strategy sometimes you catch more with honey than with vinegar and sometimes as in the post office the strategy we were adapting is full frontal assault so far repelled so anyway we are trying to think constructively about how we as a community can get the most out of these ideas and your ideas on that are useful as well so you have a church you didn't have to someone know about it we won't use the disease spreading mechanism analogy here I'm Judy Walk and I'm a member of Christ Episcopal Church which as you know got a a lot of water again and what I'm learning is that you know the people in the church who are trying to figure out how the hell to build back better safer are learning a ton of stuff and I don't know that anybody is sharing what they're learning from wherever they're cleaning this kind of either professional engineers or funding agencies or pick one I wish there was a way to be sure that those people were telling each other what they've learned about the different kinds of wet and dry and all this stuff that you know to find a way that people who are trying to take action now have access to what other people have found out I have no idea how you would do that but it feels like there's an opportunity it has a good point can we connect through you? oh Katie can we connect through you to people well I'm sort of related because we talked about this in the first session but I think education is so important because not everybody is on the same wavelength as everybody in here and we have to understand that the water is going to come again and things are going to get worse and we have to be preemptive and it doesn't have to be bad it can be good oops sorry I love what you just said if you're okay to pivot a little bit what do folks think we need if we were to launch an education effort and we had some themes that we're going to talk about to the public what do you think we would need to cover what are the urgent ones that are misunderstood I'm Jared Duvall and just to build on the last two comments I wonder about some sessions that would kind of just present options for adapting existing buildings for flood resilience it seems to me like there's I don't know exactly what it is to play that it is probably almost like some kind of decision tree if X, Y, or Z is present then you know hear your options if not then hear your options and I just don't have a really good sense of how much that is known by the different building owners downtown I feel like what you were saying Judy like I'm aware of similar conversations happening at the Unitarian Church of Montpelier and I think that these are probably happening building by building to which common themes or common learning could be shared I think could be really helpful so that everybody doesn't feel like they're doing it on their own and you know maybe that's bringing in experts or sharing local expertise in kind of open forums but it feels like there's enough different options for what does this what does adapt existing buildings for flood resilience mean that it's worth digging deeper and sharing more knowledge it's out of note here I've been talking to a bunch of the downtown merchants and a couple of realities that I think we have to face is that many of the merchants are not owners of the buildings they're renters okay and one of the things that has been striking me in the conversations I've had is that I can't do this again so if there is a prospect of another flood it's not that what the building owner does it is this location of the merchants which creates a whole other category of learning and support so I think some part of this has to be around creating a sense within our merchant community that there is some kind of support being built toward the future of our downtown and that it is there for them as if another disaster strikes because without them like I said a lot of them say I can't do this I'm not going to bother so I think we have to take that to account so whether the building gets fixed or not and some of them are far from being fixed yet we've got a problem Paul Castello I think there's a like a fundamental contradiction of this kind of work between being holistic and looking for a big long-term solution that solves everything or being pragmatic and identifying things that will move the ball and I think that one question I have is the potential of the commission to provide leadership point to point rather than top down we're going to teach you folks what's good for the future demonstration projects like has there been a conversation with Efficiency Vermont about getting in line and trying to replace building by building heating systems and demonstrating that Montpelier is a place with modeling solutions for upstairs electric heating that's both climate friendly and also not destroyed in whatever next but happens like things like that tell a story that's good for the community to hear and also others but also looking at things like the doors and is there a pattern for doors we're not going to move everybody out we don't have top down control over the decisions of people who put sweat equity for generations in these buildings and who have livelihood we have the opportunity to set out markers that show some of the ways forward and also argue with the different powers that be one of the conversations in the other room was what's the authority of the commission and there's no authority to the commission but in truth because the whole community's had opportunities to throw their ideas at the commission the commission is empowered to represent the community and take bold direct sequential action that builds momentum momentum's more valuable than a final solution because the final solutions are going to take a generation that's my guy thank you Ken and then Ken Jones and I was at the agency of commerce post Irene so when a lot of this resilience discussion happened there I got to participate and one of the themes I think that's very important is resilience is a lot easier if there's an economic strength before the event so I would and I know this is an opinion that not everyone shares but maybe there's enough momentum we need to support economic growth in our downtown or else our economic growth will happen along the interstate exits because where's the next hotel going to be built there's one in Randolph right off the exit talking about expanding up in Berlin because we didn't want one here and if we keep doing that we're not going to have the economic strength in our downtown so when the next event doesn't happen it will be much easier for those businesses to leave so I really would like us to rally our support to say we may be building buildings that don't appeal to everyone's aesthetic sense but if it draws in more people if it draws in more activity that's what we're going to need to get through the next flood I think to not to Peter walk, by the way I think the thing that I think about a lot post flood is a bit of a counterpoint to what you said Dan not to disagree with you that it's that our merchants are facing a somber reality but in the immediate aftermath I was surprised anybody wanted to come back and I give a two shout out to Katie Trapp over there for rallying our businesses to understand what they needed to do to have good information because it was in short supply but the fact that everybody wanted to come back and their community rallied around them is one of our strengths and we need to figure out how to build from that you just talked a little about infrastructure and improvements after the flood I think it's worth mentioning that the city planning department I think did a pretty good job of doubling down and making sure that existing regulations said that substantial improvements required that when that equipment was replaced it goes to be above it's protected and I don't know that that hasn't necessarily maybe been as much of a public announcement that that was that that happened and that in the education front maybe something that can happen is like okay well this commission speaking for like those that are concerned about resiliency in Montpelier saying like here are the guidance here if there's a landowner that maybe didn't do this this is happening this is what the rules say this is what it has to be that's our bell, we'll hear from Barbara and there are pieces of paper out on tables there are post-it notes, lots of opportunities to give us input please don't be shy, I know not everybody wants to speak up in front of the group yeah, in reference to that I'm involved with the Unitarian Church in Montpelier and we went to replace our heating system and the city told us that it had to be two feet above the flood plain which was the flood level was our lower floor and so we kind of went berserk and had to raise all kinds of money and we were like you know really grumbling about it and then the flood came and our mechanical system was fine so you know you can definitely see the value of it I want to applaud the throwing and catching in here really good this season's about news team here thank you all for coming thank you


See we are discussing diagonalizability in today's lecture we will derive one necessary and sufficient condition for diagonalizability of a linear operator. So let us go back and look at this problem diagonalizability of a linear operator T this means that I can write down the matrix of T relative to some basis as a diagonal matrix okay in writing down this diagonal matrix let me assume that by the way is it clear that the eigenvalues of this diagonal matrix must be the eigenvalues of T the eigenvalues of the diagonal matrix so what is the meaning of saying that T is diagonal this means T ui so I am just writing down the equation that I gave yesterday instead of alpha let me write lambda i ui these are the numbers that will figure here these lambda i's are the numbers that will figure here okay. Now some of we have seen example the second example where the eigenvalue repeats so some of these could repeat so what I will do is write down the distinct eigenvalues and take care of multiplicity when I write down the diagonal matrix in other words let me say lambda 1, lambda 2, etc lambda k be the distinct eigenvalues be the distinct eigenvalues of T with multiplicities I will use this notation different from what I used yesterday in the last lecture with multiplicities let me say D 1, D 2, etc D k that means lambda 1 comes as an eigenvalue of the operator T D 1 times in other words the characteristic equation that is characteristic polynomial is determinant of T minus lambda i is 0 characteristic polynomial is determinant of T minus lambda i characteristic equation is determinant of T minus lambda i equal to 0 this is a polynomial of degree n when I write down this what I mean is that lambda 1 appears D 1 times as a root of that characteristic equation lambda 2 appears D 2 times etc lambda k appears D k times okay so what is clear is that D 1 plus D 2 etc plus D k is equal to n the degree of the polynomial okay if this is the case then this can be rewritten then I can write this matrix diagonal matrix as lambda 1 now appears D 1 times so the first block will have lambda 1 appearing D 1 times is it clear that I can write it as lambda 1 i 1 lambda 2 i 2 etc lambda k i k all other entries are of course 0 with the convention that i 1 is the identity matrix of order D 1 cross D 1 i 2 is identity matrix of order D 2 cross D 2 etc i k is identity matrix of order D k okay so let me confirm where i i is the identity matrix of order D i so I can first this is the first observation the diagonal matrix D can be written in this manner after listing the distinct Eigen values so let us observe that the statement that I made just now D 1 plus D 2 plus etc D k must be the dimension of the space that I started with I will always assume the dimension to be n okay in the rest of the discussion okay if this is the case what is the characteristic polynomial of T the explicit formula for the matrix of T has been written down we know that the characteristic polynomial does not change when I change the basis okay because it is basically the determinant of some matrix which does not change under change of basis okay so the characteristic polynomial so I am using P for the characteristic polynomial I will use P here after for the characteristic polynomial so the characteristic polynomial of this operator T can be written as it will be lambda will be a variable lambda 1 lambda 2 etc are the zeros so I have lambda minus lambda 1 to the D 1 into lambda minus lambda 2 D 2 etc lambda minus lambda k to the D k is it clear that this is the characteristic polynomial of T okay see in general this cannot be done I am assuming that if T I am assuming that T is diagonalizable in that case I can do this for example look at the operator first operator that we considered yesterday the rotation matrix when the angle is 19 the characteristic polynomial there is lambda square plus 1 okay and lambda square plus 1 cannot be factorized as lambda minus lambda 1 into lambda minus lambda 2 for lambda 1 lambda 2 real because this is an irreducible polynomial over the polynomial ring f x R x okay irreducible polynomial over R lambda square plus 1 is one example we cannot factorize there that is an example of an operator which is not diagonalizable so this cannot be done always if it is diagonalizable then this can be done if the operator is diagonalizable then the characteristic polynomial can be factorized into products of powers of linear factors let me mention there is a product of powers of linear factors okay okay so this is one information that the characteristic polynomial is a product of powers of linear factors is one information I also have the other information look at look at the eigens look at the dimension of the subspace null space of T minus lambda I I I want to calculate the dimension of the subspace null space of T minus lambda I I I I want to calculate the nullity of T minus lambda I okay can you tell me what this is what is this subspace first see I want T minus lambda I I I have T here I can treat like matrices for being specific let us take lambda I to be lambda 1 I is 1 lambda 1 T minus lambda 1 I this first block is 0 all the other entries will remain why because they are distinct this will be lambda 2 minus lambda 1 times I 2 this will be lambda k minus lambda 1 times I k I am doing T minus lambda 1 I so these entries will not be 0 distinct eigenvalues so lambda 1 will not be equal to lambda 2 etc lambda k but this block is 0 I want the set of all solutions of the matrix equation let us say some some A x equal to 0 where A is a matrix whose first block is 0 all other diagonal entries are not 0 what is the dimension of the solution space is it clear it is D I the rank of T minus lambda 1 I will be the rank coming out of these non-zero entries so nullity corresponds to just this so null the nullity of T minus lambda I is D I please check this so what is it clear first the null space has this dimension by the way what is the null space of T minus lambda 1 I can you see that it is the it is the eigen space corresponding to the eigenvalue lambda I this I did not mention in the last lecture this is the set of actually the space the set of all eigenvectors of T corresponding to the eigenvalue lambda I this is the set of all eigenvectors of T corresponding to the eigenvalue lambda I first observation this is a subspace I did not make this point yesterday the set of all eigenvectors corresponding to an eigenvalue forms a subspace because it is a null space of a certain linear transformation null space of T minus lambda I so this subspace has dimension D I remember there are there are two objects coming here one object is a polynomial the other one is the dimension of some subspace if T is diagonalizable these two numbers are the same this is an important observation because we will see that you can go back to example 2 example 3 no example 2 we discussed only 2 examples you can go back to example 2 look at the dimension of the subspace corresponding to the eigenvalue 1 the dimension is 1 there whereas the multiplicity of 1 as an eigenvalue is 2 in general these are different okay for diagonalizability to crucial that these two numbers are the same okay this is the second important point first point is that the characteristic polynomial is can be written as a product of powers of linear factors second fact is that the dimension the number of times lambda I appears as an eigenvalue of A that is D I that number is the same as the dimension of the eigen space corresponding to the eigenvalue lambda I okay okay let us proceed this is important okay as you will see this is important for diagonalizability okay as I told you I want to look at a characterization let me give you 1 or 2 results before I prove that result first I want to demonstrate the following suppose T is an operator on a finite dimensional vector space V and F of lambda okay F of T I will use small t okay let me just say F be a polynomial over F V is a vector space over F F is a polynomial over F by which I mean that this polynomial has its coefficients coming from F okay so if F is R this is a real polynomial if T X equals lambda X for lambda in the underlying field then F T of X is F lambda of X we will need this little result then F T of X is F lambda of X what this means is that if lambda is an eigenvalue for the operator T and if X is eigenvector then look at F of T F of T is also a linear operator see little f is a polynomial F of T is another linear operator for this linear operator we want to show that F of lambda is an eigenvalue with the same eigenvector X with the same eigenvector X okay prove straight forward you just write down F of T let us say it is A naught plus A 1 T plus A 2 T square etc let us say A S T to the S where the coefficients come from the underlying field then what is F of capital T wherever little T comes I must replace it by capital T the first term it is T power 0 that is identity operator okay so my F of T is this polynomial A naught I plus A 1 capital T plus A 2 capital T square etc A S T to the S remember T is an operator on V it is a linear transformation from V to itself so this T square T cube etc makes sense composition this is F of T what do I want to verify verify this so let me write F T X this will be A naught I X plus A 1 T X plus A 2 T square X plus etc I need to calculate each term okay but it is easy to see that T okay let us calculate T square X for example T square X is T of T of X T of X is lambda X so this is T of lambda X this is lambda T of X again T of X equals lambda X so this is lambda square X so T square X is lambda square X by induction T power R X is equal to lambda power R X so this can be rewritten okay so this is a simple result so if you look at F T X it is A naught X plus A 1 lambda X plus A 2 lambda square X plus etc plus A S lambda to the S X X is a vector take that outside all the others are numbers coefficients A naught plus A 1 lambda plus etc plus A S lambda to the S into X this is a this is a number coming from the field but this is precisely F of lambda instead of T I have lambda so this is F of lambda X okay so remember that F T is a polynomial in T if lambda is an eigenvalue and X is a corresponding eigenvector then we have found out an eigenvalue for F of T that is F lambda for the same eigenvector X okay okay so that is a simple computation I want one more result before I prove the main theorem so the framework is the same as before lambda 1 lambda etc lambda k are the distinct eigenvalues of a linear operator T on a finite dimensional vector space V let me say that W I let W I be the eigenspace corresponding to the eigenvalue lambda I so I have these k eigenspaces corresponding to the eigenvalue lambda 1 etc lambda k now these eigenspaces are subspaces I can talk about the sum of these subspaces let W equal to W 1 plus W 2 etc plus W k I take the sum of these k eigenspaces now remember that dimension of W 1 plus W 2 in general is not equal to dimension W 1 plus dimension W 2 you need to remove the dimension of W 1 intersection W 2 okay but in this case the dimensions add up in this case the dimension will add up that happens if the subspaces are eigen space that is what we are trying to do here so what is the conclusion then dimension W is dimension W 1 plus dimension W 2 etc in particular what this means is that eigenvectors corresponding to distinct eigenvalues are linearly independent okay this is this statement is encoded in this eigenvectors corresponding to distinct eigenvalues are linearly independent okay so let us prove this then we will use this in characterizing diagonalizability all that I will do is take a basis for W 1 take a basis for W 2 etc basis for W k show that the union is a basis for the sum W is that okay I will take a basis for W 1 base for W 2 etc W k take the union I will show that that is a basis for W then it follows that dimension of W is the number of elements in B 1 plus the number of elements in B 2 etc number of elements in B k that is precisely this okay so let us write down now basis explicitly let B 1 equal for the subspace W 1 corresponding to lambda 1 so B 1 I will call it U 1 1 U 1 2 etc U 1 L 1 the first subscript denotes that it corresponds to that eigenvalue first subscript denotes it corresponds to the eigenvalue lambda 1 okay so this corresponds to eigenvalue lambda 1 B 2 is U 2 1 U 2 2 etc U 2 L 2 this corresponds to lambda 2 etc B k U k 1 U k 2 etc U k L k this corresponds to the eigenvalue lambda k let these be basis ordered basis for W 1 W 2 etc W k I must show that I must show that this is a basis for W okay spanning set we will disposed of quickly linear independence is what is a little difficult in this problem little more involved than the other one spanning set I want to show that this union this is a spanning set for W okay take anything in W little W that is W 1 plus W 2 etc W k each W 1 in turn can be written in terms of these so it is clear that this is a spanning set clearly this is this spans W linear independence is see to show that this is a basis for W I must show that it is a spanning set and it is a linearly independent set the fact that this is a spanning set is straight forward for the following reason take any little W in W then by definition that W is some W 1 plus W 2 etc W k but look at W 1 the first term that is a linear combination of these etc so this W is a linear combination of these vectors that is those vectors in script B and so this spans W linear independence consider so we need to show linear independence so we need to consider a combination so consider alpha 1 1 U 1 1 plus alpha 1 2 U 1 2 I choose the scalars also according to the superscripts etc alpha 1 L 1 U 1 L 1 plus alpha 2 1 U 2 1 alpha 2 2 U 2 2 plus etc plus alpha 2 L 2 U 2 L 2 plus etc last term alpha k 1 U k 1 alpha k 2 U k 2 etc alpha k what is that last one L k U k L k suppose this is 0 I must show that each of the scalars is 0 okay consider this combination I must show that each scalar is 0 it would then follow that these vectors U 1 1 U 1 2 etc U 1 L 1 U 2 1 U 2 2 etc U 2 L 2 U k 1 U k 2 etc U k L k they are linearly independent okay let us this is this vector is a 0 vector f of t easily take any polynomial for any polynomial f of t I will consider f of capital T f of capital T is also linear I will apply f of capital T on this of this whole thing okay alpha 1 1 U 1 1 plus alpha 1 2 U 1 2 etc alpha 1 L 1 U 1 L 1 this time I will just write down okay does not matter alpha 2 1 U 2 1 etc alpha 2 L 2 U 2 L 2 plus etc the last one the last one is alpha k 1 U k 1 plus etc plus alpha k L k U k L k f of t is also a linear operator so any linear operator has a property that it is action on the 0 vector this is a 0 vector so 0 is this f of t is linear because t is linear I will apply this to each term okay so can you see that the first term will be alpha 1 1 f t U 1 1 use the previous lemma it is f of remember U 1 1 is an eigenvector corresponding to the eigenvalue lambda 1 all these U 1 1 U 1 2 etc U 1 L 1 see they come from they come from the eigen space corresponding to the eigenvalue lambda 1 so each of this is an eigenvector corresponding to the eigenvalue lambda 1 apply the previous lemma f t of x equals f of lambda x if lambda is the eigenvalue so the first set of terms is it clear that it is alpha 1 1 f lambda 1 U 1 1 plus f lambda 1 U 1 2 etc U 1 L 1 first terms will go with f lambda 1 plus the second ones will go with f lambda 2 the last one will go with f lambda k do you agree with this I forgot the constants here also okay is it clear the first set of terms coming from this bracket the first one yeah the first set of terms here they will go with f lambda 1 because each of those vectors U 1 1 U 1 2 etc U 1 L 1 each of those vectors is an eigenvector corresponding to the eigenvalue lambda 1 and I am appealing to the previous lemma okay this is true for any polynomial f this is true for any polynomial f I will make now particular choices of f and then show that this is 0 another choice I will show this is 0 etc okay suppose I have one choice for which the second set of terms third set of terms etc the last set of terms vanish this only remains does it follow that those those scalars those scalars will be 0 okay I will choose that polynomial in such a way that this is not 0 so it will follow that alpha 1 1 alpha 1 2 etc alpha 1 L 1 is 0 apply the next polynomial I will choose it in such a way that f lambda 2 is not 0 it will follow that these coefficients are 0 etc I will go back and substitute into this equation I am sorry I want to conclude each scalar is 0 so it will follow that these are independent okay now what is that polynomial for the first one I will do it for the first one the rest is similar for the first one I will choose f of t to be say I want these to be 0 I want a polynomial I want the polynomial to have lambda 2 etc lambda k to be 0 give me one choice t minus lambda 2 t minus lambda 3 etc if you choose this then f of lambda 1 f of lambda 1 is lambda 1 minus lambda 2 lambda 1 minus lambda 3 etc product lambda minus lambda k these are distinct so that is not 0 okay so f of lambda 1 is not 0 f of lambda 1 is not 0 but f of lambda i equal to 0 for all i greater than or equal to 2. So I will take this polynomial apply it to this equation then I get okay I take this polynomial apply this polynomial to this equation then I get the second set of terms are 0 etc I have only these terms remaining I can write it as f lambda 1 outside into the rest of them alpha 1 1 u 1 1 etc alpha 1 k l 1 u 1 l only the first set of terms will remain f lambda 1 is not 0 so this can be cancelled look at the rest that is 0 but these are linearly independent right they form a basis for w 1 these are linearly independent from this it follows that the first set of coefficients alpha 1 1 alpha 1 2 etc alpha 1 l 1 first set of coefficients must all be 0 so you apply the second polynomial which is t minus lambda 1 into t minus lambda 3 t minus lambda 4 etc t minus lambda k then you can show the second set of coefficients 0 etc so it follows that each of these scalars started with this equation follows that each of the scalars is 0 and so these vectors are linearly independent so this is the basis for the space w okay as I mentioned earlier in particular this means that Eigen vectors corresponding to distinct Eigen values are linearly independent we have proved something more okay let me prove this theorem then theorem that characterizes diagonalizability the framework is as before t is a linear operator on a finite dimensional vector space with dimension n over f v is defined over f I have lambda 1 lambda 2 etc lambda k as the distinct Eigen values let w i be the Eigen space corresponding to the Eigen value lambda i then the following statements are equivalent I will give two conditions that are necessary and sufficient for t to be diagonalizable first statement is t is diagonalizable second is a condition involving the representation of the characteristic polynomial characteristic polynomial I am denoting by p I will write p of lambda it is lambda minus lambda 1 to the d 1 lambda minus lambda 2 to the d 2 etc lambda minus lambda k to the d k where d 1 plus d 2 plus etc plus d k equals n I am assuming that n is a dimension of the space v the last condition is in terms of sums of the Eigen spaces if you look at the sums of the Eigen spaces call it w then this w the subspace is the whole of the space v this is the last condition we have already observed that a implies b if t is diagonalizable then we have seen that the characteristic polynomial has this form from b implies c proof a implies b already observed b implies c follows because okay you can set w to be the sum of these subspaces then by the previous lemma it follows that dimension of w is summation i equals 1 to n dimension w i this comes from the previous lemma because these are Eigen spaces corresponding to distinct Eigen values but what is the condition on look at condition b the condition on these numbers d 1 d 2 etc d k is that their sum is n so dimension w is summation i equals 1 to n d i that is n so I was 1 to k there are k subspaces yeah so this sum is n so but we know that the sum of two subspaces again a subspace w is a subspace of v having the same dimension so w must be the whole of v that statement c the sum is equal to w w is equal to v we have shown so b implies c also holds c implies a c implies a follows from the definition of what we mean by diagonalizability is that clear I want to show t is diagonalizable that is I want to show that there is a basis script b of v such that each vector from b is an Eigen vector of t that is diagonalizability we saw this yesterday t is diagonalizable if and only if there is a basis b for v each of whose vector is an Eigen vector for t I know that v is the direct sum of is a sum of these subspaces look at the construction that we did earlier take a basis b 1 for w 1 b 2 for w 2 etc b k for w k each of the basis has a property that each of these basis for the subspaces b 1 b 2 etc b k have the property that their elements the vectors in b 1 for instance is an Eigen vector corresponding to lambda 1 the vectors in b 2 are Eigen vectors corresponding to the Eigen value lambda 2 etc the combination is a basis for v the union of these basis for the subspaces is a basis for v and so each vector of this basis is an Eigen vector of v is an Eigen vector of t so t is diagonalizable so can I just say c implies a follows from really the definition which we saw yesterday really the definition that there is a basis b for v each of whose vectors is an Eigen vector for t okay so this is one characterization that is necessary sufficient condition for t to be diagonalizable one is that the characteristic polynomial can be written as a product of powers of linear factors second is that the whole space v can be written as a sum of these subspaces the subspaces being the Eigen spaces okay this is one characterization we will also look at another characterization involving the so called minimal polynomials okay that I will do in the next one or two lectures but before I conclude I want to at least mention what is a minimal polynomial we want to know when precisely a linear transformation is diagonalizable okay one answer has been given here two answers really look at the Eigen spaces take the sum verify if that is the whole space v the other thing is look at the characteristic polynomial verify if it is a product of powers of linear factors then you know that t is diagonalizable there is another answer as I told you which comes in terms of the minimal polynomial okay so let me at least give the definition of the minimal polynomial remember that we are looking at an operator a single linear operator t and we are analyzing the operator t in this study what is important is to identify classes of polynomials which have the property identify classes of polynomials let us say f of t such that f of capital T is 0 what are all polynomials f of t such that f of capital T is 0 now why is this statement true that will be clear only a little later we will get the connection between minimal polynomial and the characteristic polynomial then it will be clear as to why these polynomials are important okay but let me at least give the concept of the minimal polynomial coming from this okay remember f is a polynomial its coefficients come from the underlying field okay now if you look at f t call that script a look at f t call that script a then see what is this f t f t is a set of all it is it has an algebraic structure it is a set of all polynomials in a single variable t in a single real variable t I should actually write f but for the sake of convenience I am writing r single real variable t so I should actually write r of t okay let me go back and change this to f so it is either for our discussion let us say it is a set of all real polynomials over a single variable t single real variable t the coefficients are real the variable t is also real I am using script a for that it is what is called as an algebra you know that it is a Euclidean domain set of all polynomials it is a Euclidean domain it is a commutative ring where you can do Euclidean algorithm it is a commutative ring where Euclidean algorithm can be applied it is an Euclidean domain which has a property that you know the concept of an ideal concept of an ideal in a sub ring okay does not matter if you do not know you will learn it this semester sometime what can be shown is that an ideal is a sub ring of see this is what is called as an algebra an algebra is an algebraic structure where you can do multiplication of vectors okay so an algebra is something more than a vector space where there is also a possibility of multiplying vectors now multiplication here is multiplication of polynomials multiplication of polynomials you know term by term multiplication of polynomials is term by term so one could do multiplication of polynomials it also has one or two little axioms but let us not worry about that this is an algebra this is a Euclidean domain where I can do where product of vectors make sense in such a Euclidean domain the notion of an ideal comes an ideal is a sub ring an ideal is a sub ring which has a property that if I take an element from the ideal and an element from outside the ideal that is f of t then the product will belong to the ideal okay then the product will belong to the ideal ideals in a Euclidean domain have the property that they are generated by a single unique polynomial ideals in a Euclidean domain are characterized by the property that they are generated by a single unique element which means anything in the ideal is a multiple of a specific polynomial anything in an ideal given an ideal anything in that ideal is a multiple of a unique element coming from that ideal okay I need this property one can also do without this property for the minimal polynomial I wanted to find the minimal polynomial first of all the question is given a linear transformation t does there exist a polynomial f such that f of t is 0 okay I will give two answers one for a specific example linear transformation matrices they are equivalent so I will take this matrix 1 1 1 1 I want you to consider this polynomial f of t equals t square minus 2 t f of t is t square minus 2 t how did I get this at the end of the next two lectures you will also be able to write down such polynomials t square minus 2 t then you can verify that f of a is a 0 matrix a square equal to 2 a a square is equal to 2 a for this matrix that is the reason why I chose t square minus 2 t then f of a equal to 0 so given a linear transformation this makes sense the question does there exist a polynomial f such that f t equal to 0 makes sense I have given one example I will actually prove it I will actually prove that given a linear transformation on a finite dimensional vector space there is a polynomial which has a property that f of capital T equal to 0 tomorrow and then maybe define the minimal polynomial and how it is related to the notion of diagonalizability okay let me stop here.


I'm Omar Khazraei. I am a fellow at Berkman Klein Center for Eternals Society and I'm also an assistant professor of information at Kansas State's School of Information in Ohio. Broadly speaking I'm interested in how human collectives use information technology to achieve their collective goals. I look at the two levels. At one level I look at very large collectives, how they use information technology for social transformations. For example, how activists use information technology to challenge authorities. Another level I'm looking at very tightly connected communities, what we call them communities of practice, how they use information technology to produce knowledge. At Berkman Klein Center I am looking to understand how we can theorize the dynamic of evolution of the tools and methods that activists use to challenge authorities. On a personal side I'm Iranian and I have seen a lot of transformations in recent years happening in Iranian society. We've seen that a very young population, educated population, use information technology to progress the state of society. We are living in an era that the pace of technology changes and transmits. It's so high that some of the people become anxious about what's the impact of technology on our society. It's very important to see whether it helps us to improve our society or not. I think that's how it's important for the average person to see in many contexts such as repressive environments, whether the use of information technology can be a force shifting the balance towards a more just and progressive society or it might give more tools for oppressive governments to repress and restrict human freedom.