Given two matrices, [tex] A [/tex] and [tex] B.[/tex] Is the following statement true?
[tex] A [/tex] is similar to [tex] B [/tex] [tex] \Longleftrightarrow [/tex] [tex] A^k [/tex] is similar to [tex] B^k. [/tex]
By definition, A and B are similar if there exists an invertible matrix P such that B = P-1AP.
The Attempt at a Solution
Clearly, the [tex] \Rightarrow [/tex] portion of the statement holds. For example, B2 = (P-1AP)(P-1AP) = P-1A(PP-1)AP = P-1A2P.
However, I am not certain about the statement in the reverse direction. I haven't spent a terrible amount of time on it, but I can't think of any counterexamples straight off the top of my head.
Any hints or suggestions? Thanks for your time.