Similarity of Diagonalizable Matrices

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Homework Statement


A = {{2,1,0}{0,-2,1}{0,0,1}} and B = {{3,2,-5}{1,2,-1}{2,2,-4}}
Show that A and B are similar by showing that they are similar to the same diagonal matrix. Then find an invertible matrix P such that P-1AP=B

Homework Equations


P-1AP=B, or AP=PB

The Attempt at a Solution


I found the eigenvalues of both matrices to be 2, -2 and 1. I have created the matrix D={{2,0,0}{0,-2,0}{0,0,1}}. But I don't know where to go from there. Any help would be greatly appreciated.
 
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flybynight said:

Homework Statement


A = {{2,1,0}{0,-2,1}{0,0,1}} and B = {{3,2,-5}{1,2,-1}{2,2,-4}}
Show that A and B are similar by showing that they are similar to the same diagonal matrix. Then find an invertible matrix P such that P-1AP=B

Homework Equations


P-1AP=B, or AP=PB

The Attempt at a Solution


I found the eigenvalues of both matrices to be 2, -2 and 1. I have created the matrix D={{2,0,0}{0,-2,0}{0,0,1}}. But I don't know where to go from there. Any help would be greatly appreciated.

Do you know how to diagonalize A itself, in other words how to build P using the eigenvectors of A to get P-1AP=D?
 
Yes, I know how to diagonalize the matrices.
 
Do you know how similarity matrices work? What linear transformation do they represent?
 
flybynight said:
Yes, I know how to diagonalize the matrices.

Well, if you can make P-1AP = D and Q-1BQ = D what is the relationship between A and B?
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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