Okay, Actually yesterday I had 10 minutes and I forgot. I wanted to do one example, which is similar to some of the questions which you get in the prelim sometimes on Jordan canonical form, which is how much information there is about the Jordan canonical form. If you know the characteristic polynomial and the minimal polynomial. Okay, it's something very elementary. But let us do an example and it's in the notes. So I will just do that. So suppose V is finite dimensional vector space and T is an LP. You can think of it as a matrix if you like. Say if the minimal polynomial of T is MX, I just made up some pink. The characteristic polynomial p of x, sorry, got mixed up is 23. And the characteristic polynomial is let me make this bigger. Okay? This is different from the one in the notes. Okay? What are the possible Jordan canonical forms? Okay? So basically it is saying, so it can actually, if I give you the minimal polynomial, the characteristic polynomial, can you determine the Jordan canonical form? And the answer is no. Which means there are many different T is not even similar. Which could have the same minimal polynomial and the same characteristic polynomial. And teeth which aren't even similar. Okay? The Jordan canonical form determines up to similarity. But minimal polynomial and characters. That's very little information. It doesn't determine the Jordan form events. Okay, so let's see what can be determined. So let's say firstly, okay, first question. What is the dimension of V? What do you think? Yeah, I didn't tell you muted. So we have the characteristic polynomial has dimension up to 10, correct? Exactly, right. That is clear. And then what are these numbers determining 3 and 7. What can be determined for the characteristic polynomial? Is this the question sets me, yeah, go ahead. So maybe the general eigen, eigenspace this, right? Okay. Before that, okay. What are the eigenvalues? 7979, right? Yeah, 79, by the way. So the minimal polynomial and the characteristic polynomial, both will have the same roots. Distinct roots to test out the powers will be different. Okay, and then I just had dementia enough. So dimension of G7 Is three. Dimension of g nine is seven. Okay? And then time, is that. Okay? So that's about dementia. Okay. So now what about the so we're looking at possible Jordan forms, right? So the Jordan form of D will have a block for the eigenvalues and a block for the eigenvalue nights that it's clear. And the total is 10 here. 10 by 10. What's the size of the block for seven? B, seven. Good. Am I today you're the answer. I'm not sure. So so it's from the diamond, the T7 size. Remember the block of seven comes from the operator P and G seven determinants to block 17 and 19. Block nine. So this will be p by p and this will be seven by seven. Now we, okay, so far. All right, so now let's look. I've just tried to figure it out. Okay? So by the way, all the information in the characteristic polynomial that's been used. There's nothing more about the characteristic polynomial which will be helpful to us now. The characteristic polynomial that has to come in the whole size of the matrix and the size of each blocks, that's it, and the eigenvalues. So this is where the minimal polynomial x, which is, sorry, this is cubed. So let's work on each one separately. Okay, remember this is a three by three matrix. Okay? What does the information in the minimal polynomial now? So I have m x minus seven squared or 9 cubed. So let us focus on the 7th. So steep and you want to tell me what's the information about the eigenvalue 7 in the minimal polynomial? I don't know. Okay. You haven't had time not don't worry. I'm just curious. Okay. So random, you want to tell me if you've had a chance to look at this material? I mean, I guess my guess would be something with the experience they get appears twice, nine appears three times. I'm not sure. Okay, so we're looking at lambda equals 7. So what does, what role is played by this, this power for the minimal polynomial? Anyone feel rushed and everything that you're doing, five other things. But can anyone remember what that is? It's even got a variable name associated with it. So a smaller box is J2. This sort of right? Yes. Okay. So this is the em, right? And what is the role of VM? That is, G2 is the nullspace of that is one role which I am place. Okay? And the second thing is, sorry, let me write this here. So G7 is okay. And as I was saying that the law test, this is the most important thing. In B7 is K27. These are the two important things. Okay? So this m determines the, what is the power of nl of p minus lambda gives you the T7. And a second me tells me the largest size of the Jordan matrix in the B7. Okay? So remember this is a three by three matrix. Okay? So three by three matrix, I have G27, all of its. What are the other choice? Then? Go ahead on line 17. That's the only choice because no other choice makes it three by three. No other choice. So that takes care of B7. So now let's go to Lambda in mind. So lambda is nine and the M here is this one. It's three. And B19 was 7-by-7. Yeah. Okay. So this implies, of course number 1, eigenspace that the nullspace hopes. And P can be nine. These are the two things. So let's look at possible forms. Remember seven by seven. So the first one is of course it's always got J3 nine. Okay. And that's the largest one. So can you tell me more? Tell me some choices. One choice. Say it again. Okay. I could put but that doesn't make it 7-by-7. A GFP and another correct that makes it seven. Okay. So it can have to do it in an organized fashion. Now of course all of us has the J3 nine. If it doesn't have another J3 nine, then of course, let's say it has a J2 nine. Okay, There's no J3, 90 new. So then, so that makes it five. Then I need two more. So I can either do a J2 nine or I can do a J2, J1 line. Right? So if you have a second choice, if you don't have the second, 2009 them, these are the only two choice. But you have a J2 nine. And then if you don't have a J2 nine, then you only have ones. So then you have four of these. And that's that. All right? So the minimal polynomial and the characteristic polynomial don't determine the operate up to similarity. There are still many choices. Okay? Alright, so that's only one hub. This precedes. Correct? Okay, So I'll finish that this is the end of B example and I will stop here. Stop recording here, and then continue.

Lecture 6 Jordan canonical form

From Rakesh Rakesh January 06, 2022

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An example showing the possible Jordan forms if one knows the minimal and the characteristic polynomial.

Tags: minimal polynomialcharacteristic polynomialjordan canonical form

Department Name: Mathematical Sciences
Date Established: January 06, 2022
Appears In: Jordan Canonical Form

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