Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 1

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Matrix Theory: Find a matrix P that puts the real 4x4 matrix A = [2 0 0 0 \ 0 2 1 0 \ 0 0 2 0 \ 1 0 0 2 ] in Jordan Canonical Form. We show how to find a basis that gives P. The Jordan form has 2 Jordan blocks of size 2.

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Ive watched hours of videos on this topic and couldnt understand what was happening, you summed up what I was confused about in a minute and half. You just saved my ass lol

mimafabian

Each Jordan block contributes exactly one vector to the eigenvectors basis, so that gives number of blocks. The size of the largest block is determined by the largest exponent of (A-cI)^k in the minimal polynomial. For the intermediate size blocks, we consider each Vj = Null(A-cI)^j. The dimensions gives a nondecreasing sequence dj. Then dk - d(k-1) is the number of blocks of size k, d(k-1)-d(k-2) is the number of blocks of size k-1, and so on. We get block generators from Vj/Vj-1.

MathDoctorBob

How do we know that there are 2 jordan blocks? 2:16

hannygehlan

Your welcome! Each Jordan block contributes a single vector to the eigenvector basis. Here we have found only two basis eigenvectors, so only two blocks.

MathDoctorBob

Hello, you say around 3:00 that because the largest Jordan block as a 2x2 matrix with eigenvalue 2 and since we only have 2 blocks, then that means they both have to be 2x2. My question is how do you know this without calculating sum of the dimensions of the general eigenspaces dim(N(2I-A))+dim(N(2I-A)^2)? Is it not possible for there to be 3 total blocks, one 2x2 and two 1x1? If not, why? Is it because the eigenvalue 2 is on a 2-cycle?

adamrichard

Why at 2:22 you said there is no basis of eigenvector ?

electroe

@andikac You're welcome! Foresight. If I want blocks, I need to order the vectors correctly. The columns of J tell us how A acts on basis vectors. Of course you should let the eigenvectors be v1 and v2 and see what happens when you change basis. - Bob

MathDoctorBob

2:00 why is that v1 and v3 instead of v1 and v2??

Numbermind

Thank you. Great examples and explanations.

robertjohnson

@CrystaTiBoha You're welcome! - Bob

MathDoctorBob

Thanks for your videos Dr. Bob. But can you explain why we can't have 3 jordan blocks?.
I mean the invariant factors could be, x-2, x-2, (x-2)^2 (in that order). So you will get a different jordan form. How do we eliminate this possibility and just consider (x-2)^2, (x-2)^2 as our invariant factors?

nalidbass