What is the relationship between C.D^n.F and A^n in matrix algebra?

[kasper_2211]

Homework Statement

Given 3 matrices C, D, F and another matrix A, can i say anything in general about the relationship between C.D^n.F and A^n if i know that F = C^-1 and that C.D.F = A.

Homework Equations

The Attempt at a Solution

For example,
If C.D.F = A then (C.D.F)^2 = A^2 and then C.D.F.C.D.F = A^2. Since F = C^-1 i can rewrite as C.D.D.F = A^2 and so C.D^2.F = A^2. I could use induction on n to show that C.D^n.F = A^n. The thing is i don't see what this says about the general relationship. It would just prove equality. Another way would be something like, C.D.F = A so F.C.D.F.C = F.A.C then, D = F.A.C and substituting that into C.D.F would give, C.D.F = C.F.A.C.F. Again this could be used to prove that C.D^n.F = A^n for all n. And again I'm not really interested in proving that they are equal. Is there some rule of matrix multiplication, that I'm not aware of, that could be used to describe the relationship? I'm a little lost here...

 

[fzero]

Without any extra special properties, there's no further simplification of the relationship. However, since A = C D C^-1, we know that A and D are similar matrices. That is, they refer to the same linear transformation, but in different bases. C^-1 is the matrix that determines the change of basis. In line with this interpretation, it's also simple to show that A and D have the same eigenvalues.

 

[kasper_2211]

Oh, thanks a lot for the reply. I've never heard of similar matrices before (this is my first week of linear algebra), but i looked it up and it got me going. So yeah, thanks.