Proving Similarity of Matrices

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Homework Help Overview

The discussion revolves around proving the similarity of matrices, specifically demonstrating that if matrix A is similar to matrix B and matrix B is similar to matrix C, then matrix A is also similar to matrix C. The participants are exploring the implications of the similarity definitions and the relationships between the matrices involved.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are examining the definitions of matrix similarity and questioning the implications of the relationships between the matrices. They discuss the necessary conditions for proving similarity and the properties of matrix multiplication that may apply.

Discussion Status

The discussion is active, with participants providing insights and questioning each other's understanding of the definitions and properties involved. Some guidance has been offered regarding the relationships between the matrices, and there is a collaborative effort to clarify the concepts.

Contextual Notes

Participants are considering the implications of using different invertible matrices for the similarity transformations and the necessity of finding the appropriate matrix to prove the similarity of A and C. There is an acknowledgment of the need to understand the properties of determinants in this context.

cheunchoi
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Hey,
i'm having trouble proving how two matrices are similar using:
A = PBP^-1

Given:
A is similar to B
And B is similar to C

Prove that A is Similar to C?

A = PBP^-1
B = PCP^-1

so i.e. A = PPCP^-1P^-1 ... ?

Can anyone help me?
 
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cheunchoi said:
Hey,
i'm having trouble proving how two matrices are similar using:
A = PBP^-1

Given:
A is similar to B
And B is similar to C

Prove that A is Similar to C?

A = PBP^-1
B = PCP^-1

so i.e. A = PPCP^-1P^-1 ... ?

Can anyone help me?

You have the right idea.

If "A is similar to B" means A = PBP^-1 ,
does "B is similar to C" mean B = PCP^-1 ? Or is that asking a little too much?

In light of the last question, what would "A is Similar to C" mean?

Is there some property of matrix multiplication that would be useful here?
 
Are you referring to the determinant?

So you're saying i can't apply the same formula for B is similar to C to give
B = PCP^-1?

I think A is similar to C means the size of both the matrix are the same. The numbers are arranged in such a way that their determinants are the same.
 
"A is similar to B" means "there exists an invertible square matrix P such that A = PBP^-1".

"B is similar to C" means "there exists an invertible square matrix Q such that B = QCQ^-1". (It need not be that P=Q... since this sentence knows nothing about A!)

What would "A is similar to C" mean (without referencing the previous two statements)?
 
It will be helpfull to know that Q-1P-1= (PQ)-1!
 
A similar to C is

A = RCR^-1 ?

Where R = PQ, from A = PQCP^-1Q^-1 ?

and to prove that, all i need to do is find R?
 
You need to find R^-1.
 
Thanks a lot guys! I got it now =)
 

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