Let \(n\) be a positive integer and \(F\) a field. Suppose \(A\) is an \(n \times n\) matrix over \(F\) and \(P\) is an invertible \(n \times n\) matrix over \(F\). If \(f\) is any polynomial over \(F\), prove that \[ f(P^{-1}AP) = P^{-1}f(A)P. \]
Solution
For every integer \(k\ge0\), \[ (P^{-1}AP)^k=P^{-1}A^kP. \] This follows by induction: the case \(k=0\) is immediate, and \[ (P^{-1}A^kP)(P^{-1}AP)=P^{-1}A^{k+1}P. \] Now let \[ f(x)=\sum_{k=0}^r c_kx^k. \] Then \[ \begin{aligned} f(P^{-1}AP) &=\sum_{k=0}^r c_k(P^{-1}AP)^k\\ &=\sum_{k=0}^r c_kP^{-1}A^kP\\ &=P^{-1}\left(\sum_{k=0}^r c_kA^k\right)P\\ &=P^{-1}f(A)P. \end{aligned} \]