Theorem 10.1. For any square matrix B, the following conditions are equivalent:(1) limk→∞ Bk = 0,(2) lim. Bkv = 0, for all vectors v,(3) p(B) < 1,(4) || B|| < 1, for some subordinate matrix norm || ||.

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square matrix B, the following conditions - all vectors V,

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.4: Similarity And Diagonalization

Problem 50EQ

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Question

Theorem 10.1. For any square matrix B, the following conditions are equivalent:
(1) limk→∞ Bk = 0,
(2) lim. Bkv = 0, for all vectors v,
(3) p(B) < 1,
(4) || B|| < 1, for some subordinate matrix norm || ||.

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