1. Show that the following statements on orthonormal vectors and orthogonal matrices are true. (a) If U € Rxn and V R* are orthogonal, then so is UV. (b) If U E Rxn is orthogonal, then so is U-1. (c) If U € Rmxn satisfies UTU = In and UUT = Im, then m = n. (d) If the columns of UE R* are orthonormal, then for any ze R", we have |UT||≤|||| (this is called the Bessel's inequality). When do we have ||UT||||||?
1. Show that the following statements on orthonormal vectors and orthogonal matrices are true. (a) If U € Rxn and V R* are orthogonal, then so is UV. (b) If U E Rxn is orthogonal, then so is U-1. (c) If U € Rmxn satisfies UTU = In and UUT = Im, then m = n. (d) If the columns of UE R* are orthonormal, then for any ze R", we have |UT||≤|||| (this is called the Bessel's inequality). When do we have ||UT||||||?
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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Parts A and B
Transcribed Image Text:1. Show that the following statements on orthonormal vectors and orthogonal matrices are true.
(a) If U € Rnxn and V € Rnxn are orthogonal, then so is UV.
(b) If U E Rxn is orthogonal, then so is U-1
(c) If U € Rmxn satisfies UTU = In and UUT = Im, then m = n.
(d) If the columns of UE R* are orthonormal, then for any ze R", we have |UT||≤|||| (this
is called the Bessel's inequality). When do we have ||UT||||||?
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