MathDB

IMC 2002 Problem 6

Source: IMC 2002

March 7, 2021

linear algebramatrix

Problem Statement

For an

n\times n

matrix with real entries let

||M||=\sup_{x\in \mathbb{R}^{n}\setminus\{0\}}\frac{||Mx||_{2}}{||x||_{2}}

, where

||\cdot||_{2}

denotes the Euclidean norm on

\mathbb{R}^{n}

. Assume that an

n\times n

matrxi

A

with real entries satisfies

||A^{k}-A^{k-1}||\leq\frac{1}{2002k}

for all positive integers

k

. Prove that

||A^{k}||\leq 2002

for all positive integers

k

.

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