# Help proving that matrices are similar

#### Jadehaan

1. Given two 4x4 matrices, A and B, I must determine if they are similar.
A=[(2,0,0,0) B=[(5,0,-4,-7)
(-4,-1,-4,0) (3,-8,15,-13)
(2,1,3,0) (2,-4,7,-7)
(-2,4,9,1)] (1,2,-5,1)]

2. A and B are similar if, A=P^(-1)BP

3. I found the eigenvalues to be 1,1,1,2 for both matrices. I also calculated their eigenvectors and eigenspaces. I am stumped as how to show that the two are similar. I know similar matrices have the same eigenvalues, but I don't think that is enough to prove similarity.

Thanks for any help,
James

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#### e(ho0n3

Are you familiar with any of the canonical forms?

#### Jadehaan

Yes. Would finding the smith normal form help at all?

#### e(ho0n3

Sure. If they both have the same Smith normal form, then they are similar.

#### Jadehaan

Thank you very much. I am having trouble calculating the Smith Normal Form by hand. We are required to show all work. I used the math program Maple to find the Smith Normal but now I have to obtain it by hand. Is there any shortcut rather than just diagonalizing (xI-A)? Can I obtain it from the charachteristic polynomial?

#### HallsofIvy

Science Advisor
Homework Helper
You say you have found the eigenvalues (which are the same for both matrices) and the corresponding eigenvectors which you don't show. If the eigenvectors (or, more generally, the eigenspaces) corresponding to each eigenvalue are the same for the two matrices, then they are similar matrices.

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