Similarity of Diagonalizable Matrices

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Homework Help Overview

The discussion revolves around the similarity of two matrices, A and B, specifically focusing on demonstrating that they are similar by finding a common diagonal matrix. The matrices involved are A = {{2,1,0},{0,-2,1},{0,0,1}} and B = {{3,2,-5},{1,2,-1},{2,2,-4}}.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the eigenvalues of both matrices, which have been found to be 2, -2, and 1. There is an attempt to construct a diagonal matrix D={{2,0,0},{0,-2,0},{0,0,1}}. Questions arise regarding the next steps in the diagonalization process and the construction of the matrix P using eigenvectors.

Discussion Status

The discussion is ongoing, with some participants expressing confidence in their ability to diagonalize the matrices. Others are probing the concept of similarity between matrices and the implications of diagonalization, indicating a productive exploration of the topic.

Contextual Notes

Participants are working within the constraints of a homework assignment that requires showing similarity through diagonalization, and there is an emphasis on understanding the relationship between the matrices involved.

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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?
 

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