27 Polynomials of a diagonable matrix

Lemma

Suppose that $A$ has $n$ distinct eigenvalues $\lambda_{1},\dots,\lambda_{n}$ . Then there exists an invertible $n\times n$ matrix $S$ such that

g(A)=S\begin{bmatrix}g(\lambda_{1})&0&0&\dots\\ 0&g(\lambda_{2})&0&\ddots\\ \vdots&\ddots&\ddots&\ddots\\ 0&\dots&0&g(\lambda_{n})\end{bmatrix}S^{-1}

for all complex polynomials $g(\lambda)$ .

Soon we’ll extend this to the functions $g(x)=\exp(tx)$ and $g(x)=1/(s-x)$ .