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.
 

Answers and Replies

  • #2
LCKurtz
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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.

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?
 
  • #3
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Yes, I know how to diagonalize the matrices.
 
  • #4
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Do you know how similarity matrices work? What linear transformation do they represent?
 
  • #5
LCKurtz
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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?
 

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