Let A be an invertible matrix and let Az = b (A+AA)(x+ Ar) = b. inequality ||AT|| A
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
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![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||
||||](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2df4d651-a502-4e9a-93b2-a531f91b18ad%2F29cbf2c9-6aca-4099-95f8-dc88938d87c4%2Fn4iig9i_processed.jpeg&w=3840&q=75)
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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