# Similarity of Diagonalizable Matrices

## 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.

LCKurtz
Homework Helper
Gold Member

## 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?

LCKurtz