(a) Let A ∈ Rn×n. Show that it holds true thatkAk∞ = max1≤i≤nXnj=1|aij |wherekAk∞ = maxx∈Rn\{0}kAxk∞kxk∞is the matrix norm induced by the vector norm kxk∞ = max1≤i≤n|xi| for x = (x1, . . . , xn)T.(b) Consider the matrix A=[1, 2; 0, 1].(i) Calculate kAk∞ and kA−1k∞.(ii) Calculate the condition number κ(A) of the matrix A with respect to the k · k∞ norm.(c) Let B ∈ Rn×n be a symmetric matrix. Show that all its eigenvalues are less than kBk∞. (a) Let A E R"Xn. Show that it holds true that|| |l = max lal1sisnj=1where| Ar||x||4|| = maxzER- \{0} |r||is the matrix norm induced by the vector norm | = max r. for r =(11,...,In)".%3DIsisn(b) Consider the matrix A=[1, 2; 0, 1].(i) Calculate ||A|| and ||A-|0.(ii) Calculate the condition number K(A) of the matrix A with respect to the || || norm.(c) Let BE R"X be a symmetric matrix. Show that all its eigenvalues are less than || B|.

We have to prove the formula for the infinite norm of a matrix A

Answered: (a) Let A € R"X. Show that it holds…

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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(a) Let A ∈ R
n×n
. Show that it holds true that
kAk∞ = max
1≤i≤n
Xn
j=1
|aij |
where
kAk∞ = max
x∈Rn\{0}
kAxk∞
kxk∞
is the matrix norm induced by the vector norm kxk∞ = max
1≤i≤n
|xi
| for x = (x1, . . . , xn)
T
.
(b) Consider the matrix A=[1, 2; 0, 1].
(i) Calculate kAk∞ and kA−1k∞.
(ii) Calculate the condition number κ(A) of the matrix A with respect to the k · k∞ norm.
(c) Let B ∈ R
n×n be a symmetric matrix. Show that all its eigenvalues are less than kBk∞.