1. ## 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!

2. ## Re: Similar matrices Q

$a=i^{-1}ai$

3. ## Re: Similar matrices Q

So I always treat S as an identity matrix?

4. ## Re: Similar matrices Q

Originally Posted by Glitch
So I always treat S as an identity matrix?
Not always; only in this case to show that A~A.

5. ## Re: Similar matrices Q

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

6. ## Re: Similar matrices Q

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.

7. ## Re: Similar matrices Q

Ahh ok, thanks again.