and D be a diagonal matrix D = diag(A₁,..., An) such that

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.3: The Gram-schmidt Process And The Qr Factorization
Problem 23EQ
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M6

Let A € M₂ (C) be a diagonalizable matrix, P be an invertible matrix,
and D be a diagonal matrix D = diag(A₁,..., An) such that
A = PDP-¹,
and let || || be a matrix norm such that
||diag(a₁,..., an) || = = max |ai|,
1≤i≤n
for every diagonal matrix. Then for every perturbation matrix AA, if we write
B₁ = {z € C | |z − λi| ≤ cond(P) ||AA||},
for every eigenvalue λ of A+ AA, we have
n
XE Ů BK
k=1
Transcribed Image Text:Let A € M₂ (C) be a diagonalizable matrix, P be an invertible matrix, and D be a diagonal matrix D = diag(A₁,..., An) such that A = PDP-¹, and let || || be a matrix norm such that ||diag(a₁,..., an) || = = max |ai|, 1≤i≤n for every diagonal matrix. Then for every perturbation matrix AA, if we write B₁ = {z € C | |z − λi| ≤ cond(P) ||AA||}, for every eigenvalue λ of A+ AA, we have n XE Ů BK k=1
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