For $A\in \mathcal{M}_n(\mathbb{C})$, define:
the spectral radius
$$
\rho(A)=\max\{|\lambda|:\lambda \mbox{ is an eigenvalue of } A\}
$$
and the norm
$$
\|A\|=\max_{|x|=1}|A(x)|
$$
where |.| is the Euclidean norm on $\mathbb{C}^n$.
Problem: Find all $A\in \mathcal{M}_n(\mathbb{C})$ such that $\rho(A)=\|A\|$.
Thank you very much!
Best Answer
This is a solved problem. A complex square matrix $A$ is called radial if its spectral radius is equal to its spectral norm $\|A\|_2=\max_{\|v\|_2=1}\|Av\|_2$. A complete characterisation of radial matrices was found in
The main result in this paper is that $A$ is radial if and only if $A$ is unitarily similar to a matrix of the form $$ \pmatrix{D&0\\ 0&L} $$ for some diagonal matrix $D$ and some lower triangular matrix $L$ such that $\|L\|_2\le\rho(A)$.