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1. Let x E C" and y E C™, with y # 0. Consider the m x n matrix defined by A = yx*. (a) Show that rank A = 1. (b) Show that ||A||2 = ||x||2||y||2- (c) Show that ||A||F = ||x||2||y||2 (which is the same as ||x||F||y||F).

1. Let x E C" and y E C™, with y # 0. Consider the m x n matrix defined by A = yx*. (a) Show that rank A = 1. (b) Show that ||A||2 = ||x||2||y||2- (c) Show that ||A||F = ||x||2||y||2 (which is the same as ||x||F||y||F).

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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SOLVE NUMBER 1

1. Let x E C and y E Cm, with y + 0. Consider the m x n matrix defined by A = yx*. (a) Show that rank A = 1. (b) Show that ||A||2 = ||x||2||y||2- (c) Show that ||A||F = ||x||2||y||2 (which is the same as ||c||F||y||F).
Transcribed Image Text:1. Let x E C and y E Cm, with y + 0. Consider the m x n matrix defined by A = yx*. (a) Show that rank A = 1. (b) Show that ||A||2 = ||x||2||y||2- (c) Show that ||A||F = ||x||2||y||2 (which is the same as ||c||F||y||F).
Expert Solution
Step 1

Given that

x∈ℂn and y∈ℂn, with y≠0.

Consider m×n matrix defined by  A=yx*.

To prove (a), (b) and (c):

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