# Similar matrices Q

• Jul 30th 2011, 03:25 AM
Glitch
Similar matrices Q
I'm a little rusty with my linear algebra, and was wondering how to prove that two matrices are similar. The question is:

Two matrices A and B are said to be similar if there exists an invertible matrix S such that $B = S^{-1}AS$. Prove that A ~ A.

So they want me to prove that $A = S^{-1}AS$, if I'm not mistaken. How do I show this? If I recall correctly, matrices are not commutative, so it's not a matter of cancelling out the S terms. Any guidance would be great!
• Jul 30th 2011, 03:32 AM
alexmahone
Re: Similar matrices Q
$a=i^{-1}ai$
• Jul 30th 2011, 03:35 AM
Glitch
Re: Similar matrices Q
So I always treat S as an identity matrix?
• Jul 30th 2011, 03:36 AM
alexmahone
Re: Similar matrices Q
Quote:

Originally Posted by Glitch
So I always treat S as an identity matrix?

Not always; only in this case to show that A~A.
• Jul 30th 2011, 03:37 AM
Glitch
Re: Similar matrices Q
Thanks. So say that I was proving A ~ B instead, how would the process differ?
• Jul 30th 2011, 03:40 AM
alexmahone
Re: Similar matrices Q
Quote:

Originally Posted by Glitch
Thanks. So say that I was proving A ~ B instead, how would the process differ?

The matrix S would depend on what A and B actually were. If A and B were different (yet similar!), we would get an S which was different from I.
• Jul 30th 2011, 03:41 AM
Glitch
Re: Similar matrices Q
Ahh ok, thanks again. :)