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Let | be any matrix norm, and let B € M₂(C) such that ||B|| < 1. (1) If || || is a subordinate matrix norm, then the matrix I + B is invertible and 1 ||(I + B)¯`'|| |≤ ₁ —-|| B|| ` (2) If a matrix of the form I + B is singular, then ||B|| ≥ 1 for every matrix norm (not necessarily subordinate).

Let | be any matrix norm, and let B € M₂(C) such that ||B|| < 1. (1) If || || is a subordinate matrix norm, then the matrix I + B is invertible and 1 ||(I + B)¯`'|| |≤ ₁ —-|| B|| ` (2) If a matrix of the form I + B is singular, then ||B|| ≥ 1 for every matrix norm (not necessarily subordinate).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let | be any matrix norm, and let B € M₂(C) such that ||B|| < 1. (1) If || || is a subordinate matrix norm, then the matrix I + B is invertible and 1 ||(I + B)¯`'|| |≤ ₁ —-|| B|| ` (2) If a matrix of the form I + B is singular, then ||B|| ≥ 1 for every matrix norm (not necessarily subordinate).
Transcribed Image Text:Let | be any matrix norm, and let B € M₂(C) such that ||B|| < 1. (1) If || || is a subordinate matrix norm, then the matrix I + B is invertible and 1 ||(I + B)¯`'|| |≤ ₁ —-|| B|| ` (2) If a matrix of the form I + B is singular, then ||B|| ≥ 1 for every matrix norm (not necessarily subordinate).
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