Proving Similarity of Matrices

[cheunchoi]

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?

 

[robphy]

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?

 

[cheunchoi]

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.

 

[robphy]

"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)?

 

[HallsofIvy]

It will be helpfull to know that Q^-1P^-1= (PQ)^-1!

 

[cheunchoi]

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?

 

[0rthodontist]

You need to find R^-1.

 

[cheunchoi]

Thanks a lot guys! I got it now =)