4.5-7 Show that the matrix norm ||A||max(||Ax||/||x||) can also be calculated as||4||max ||Ax||||x||-1calculate ||A||2, where3- 56.(Hint: A 2-vector x has 2-norm ||x||2 = 1 if and only if= cos 0, x2sin 0 for some 0.)%3D

As per the question and the hint, we will first define the matrix norm ||A|| in 2d Cartesian…

Answered: 4.5-7 Show that the matrix norm ||A||…

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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5. Show that a complex square matrix is unitary if and only if it is unitarily equivalent to a complex diagonal matrix whose diagonal entries all have absolute value one.
1 0 A = 3 -1. Compute || A||₁||A|||||||||| 0 3 (2) For any matrix A € Cnxn, show that ||4||₁ ≤|4|| (3) If |||| is a vector norm on C", show that ||~|| ≤ Σ|x₁|||e₁|| for all_x=[x₁ i=1 n where e; denotes the ith column of identity matrix of ordern. TEC,
Cholesky's Method • Check if symmetric • A = AT 11 1 11 1 Given: 1 2 2 1 2 3 AT = 1 2 2 1 2 3 A = X, + X2 + X3 = 5 X, + 2X, + 2X, = 6 X, + 2X2 + 3X3 = 8 %3D
Consider the following interpolating points for the function f(x) = x² – 2x: Xi 01 23 f(xi) 0 -1 03 (a) Use monomial basis to find P₁(x) that interpolates f(x) at x; for i = 0,1. Con- struct the Vandermonde matrix and then use the backslash \ in matlab to solve the linear system. (b) Repeat (a) to find P₂(x) that interpolates f(x) at x; for i = 0, 1, 2. (c) Repeat (a) to find P3(x) that interpolates f(x) at x, for i = 0, 1, 2, 3. (d) Compare P₂(x), P3(x) and f(x). What do you see? Can you explain it?
We can use the lemma I showed in the second images directly
[1 -1] [1 B = |2 3 [2 -1 Let A 3. 0 and C =|1 1 .1 1 -1. .1 -2] Lo 3 a) Eind A-1, if exists. b) If possible, find the matrix X such that AX + B = C. c) If possible, find a matrix Y such that YA +B = C.
7 and 5
2. (а, Sketch each of the following sets of complex numbers z that satisfy the given equations/inequalities (show your work): i. |z +3 – i| = V2 ii. |2 +3| < |2 + 1|. (b) and A? is symmetric. [Note AT denotes the transpose of the matrix]. If A is a skew-symmetric matrix then show that AAT = AT A %3|
2 2 (1) Let A Does there exist an invertible matrix P such that P-'AP = 3 1 2 0

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4.5-7 Show that the matrix norm ||A|| max(||Ax||/||x||) can also be calculated as ||4|| max ||Ax|| ||x||-1 calculate ||A||2, where 3 - 5 6. (Hint: A 2-vector x has 2-norm ||x||2 = 1 if and only if = cos 0, x2 sin 0 for some 0.) %3D