MathDB

Let

A

be a

n\times n

complex matrix whose eigenvalues have absolute value at most

1

. Prove that

\|A^n\|\le \dfrac{n}{\ln 2} \|A\|^{n-1}.

(Here

\|B\|=\sup\limits_{\|x\|\leq 1} \|Bx\|

for every

n\times n

matrix

B

and

\|x\|=\sqrt{\sum\limits_{i=1}^n |x_i|^2}

for every complex vector

x\in\mathbb{C}^n

.)
(Proposed by Ian Morris and Fedor Petrov, St. Petersburg State University)

Problem Statement