Problem 1 151025 pts a consider a matrix a in mathbbr5 times...

Problem 1 [15+10=25 pts] a) Consider a matrix $A \in \mathbb{R}^{5 \times 5}$ whose eigenvalues are: $\lambda_1 = \lambda_2 = \lambda_3 = 2$ and $\lambda_4 = \lambda_5 = -1$. Additionally, it is given that $rank(A - 2I) = 4$ and $rank(A + I) = 3$. Determine $J$ where $J$ is the Jordan form of $A$. b) Write the closed-form expressions of $e^{Jt}$ and $J^k$. Here $J$ is the Jordan matrix you obtain in part (a).

Transcript

00:01 The problem we're looking at today gives us the eigenvalues, they're just lambda 1 and lambda 2, but you don't need to know the actual values for our purposes of a 5x5 matrix, and we are given that lambda 1 is not lambda 2, so they're distinct, and that the multiplicity m1 of lambda 1 is 3, so that's how many times it's repeated, and m2 equals 2.

00:26 And from that, we need to list all the jordan canonical structures, that a matrix with these properties can have.

00:33 So basically, we are going to have our general jordan matrix.

00:42 It's going to be of the form, right? so that's just the definition of the jordan canonical matrix.

00:54 And it has zeros down here as the eigenvalues on the diagonal.

01:00 And we need to know what are we going to put here? where are ones going to be on the super diagonal, right? so how many different ways can we break this up into jordan? blocks.

01:10 So that is equivalent to saying how many different ways can we break up the number three into additive groups because we don't care about permutations.

01:18 So two plus one is going to be the same as one plus two.

01:21 So three we can break up into three two plus one one plus one and then two that's going to be two and one plus one.

01:41 So pretty simple.

01:42 Now what are these going to look like? well, i'm going to draw them all on the same matrix, but a question asks for them to list them all.

01:50 So you're going to have to draw five different matrices because we have five different options here.

01:55 But you can pause the video and copy it down if you need.

01:58 But i'm just going to do them on the matrix to save space.

02:02 So what would a matrix with three, a block of size 3 in the lambda 1s and a block of size 2, lambda 2s look like? that's this.

02:14 You see they're all in a block and sorry, no.

02:23 Yes, that's correct.

02:26 I got in my own head for a second, but yes.

02:28 So that's a block of size two and a block size three.

02:31 And then two plus one for the lambda ones is going to look like that.

02:37 And one plus one plus one is just they're all in a row.

02:40 All the super diagonal is zero.

02:43 It's essentially non -defective specifically for lambda ones.

02:47 And then we can do all the same things, but lambda 2 is going to be two different groups of 1.

02:54 So we erase that and then we have all our different permutation possibilities for lambda 1.

03:00 So that's again 3, 2 plus 1, 1 plus 1, and there we go...

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