# is it true that for any $n \times n$ matrix $A$, $Q$ where $Q$ is invertible, $(Q^{-1}AQ)^k = Q^{-1}A^kQ$?

I was trying to solve a problem that asks me to show $p(Q^{-1}AQ) = Q^{-1}p(A)Q$ where $p(t)$ is an arbitrary polynomial $a_nt^n+...+a_1t + a_0$. I am wondering whether it is true that $(Q^{-1}AQ)^k = Q^{-1}A^kQ$ since if this is true then the problem can be solved easily. I cannot seem to prove it since I am not quite sure how to deal with a product of matrices raised to a power. If it's not true then is there any other way to solve the problem?

• $A² = AA$, $A^3 = AAA$, ect. Try writing your product out for certain $k$ and see what happens. – Kaynex Jun 17 '18 at 0:49

Suppose it is true for $n$, $(QAQ^{-1})^{n+1}=(QAQ^{-1})^n(QAQ^{-1})=QA^nQ^{-1}QAQ^{-1}=QA^{n+1}Q^{-1}$ and the proof follows recursively.

• for the second equation the $(QAQ)$ term, is it supposed to be $(QAQ^{-1})$? – PsychoCom Jun 17 '18 at 1:02

You can just open the power k and write $(Q^{-1}AQ)^{k} = (Q^{-1}AQ) *(Q^{-1}AQ)*...*(Q^{-1}AQ)$ where $Q^{-1}AQ$ appears k times. It immediately gives the result after canceling all $Q^{-1}Q$s.