Let A be an invertible matrix and let Az = b (A+AA)(x+ Ar) = b. inequality ||AT|| A

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Let A be an invertible matrix and let x and x + Ar be the solutions of
Proposition
the two systems
Ar = b
(A+AA)(x + Ax) = b.
If b0, then the inequality
||AT||
||x + Ax|| cond(A) |AA||
<
|| A||
holds and is the best possible. This means that given a matriz A, there exist a vector b #0
and a matrix AA #0 for which equality holds. Furthermore, if ||AA|| is small enough (for
instance, if ||AA|| <1/ ||A¹||), we have
||AT||
||||
< cond(4) A (1+0(||AA||));
|| A||
in fact, we have
1
<cond(A)-
||AA||
|| A||
1- ||A¹||||AA||
AAT||)
||AT||
||||
Transcribed Image Text:Let A be an invertible matrix and let x and x + Ar be the solutions of Proposition the two systems Ar = b (A+AA)(x + Ax) = b. If b0, then the inequality ||AT|| ||x + Ax|| cond(A) |AA|| < || A|| holds and is the best possible. This means that given a matriz A, there exist a vector b #0 and a matrix AA #0 for which equality holds. Furthermore, if ||AA|| is small enough (for instance, if ||AA|| <1/ ||A¹||), we have ||AT|| |||| < cond(4) A (1+0(||AA||)); || A|| in fact, we have 1 <cond(A)- ||AA|| || A|| 1- ||A¹||||AA|| AAT||) ||AT|| ||||
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