$\|A\|_*=\rho(A)$ for some induced matrix norm when $A$ is diagonalizable.

Question

I'm trying to show that for $A\in \mathbb{C}^{n\times n}$ with $A$ diagonalizable, that there exists an induced matrix norm such that $\|A\|_*=\rho(A)$.

I know that $\rho(A) \leq \|A \|$, for all induced matrix norms. But I'm stuck on show the other direction, $\| A \|_* \leq \rho(A)$.

Any suggestions on how to proceed?

Answer 1

Solution Outline: Let $\|\cdot \|$ refer either to the $2$-norm or its induced matrix norm. For any invertible matrix $S$, we can define the vector norm $\|\cdot\|_S$ by $$ \|x\|_S = \|Sx\|. $$ Show that the induced norm satisfies $$ \|A\|_S = \|SAS^{-1}\|. $$ Using the definition of diagonalizability, argue that there exists a matrix $S$ such that $\|A\|_S = \rho(A)$.