Let A be an n x n symmetric matrix, let M and m denote the maximum and minimum values of the quadratic form XTAX, and denote corresponding unit eigenvectors by ul and un. The following calculations show that given any numbert between M and m, there is a unit vector x such that t = x"Ax. Verify that t = (1 – a)m + ¤M for some number a between O and 1. Then let x = V1– au, + Jauj, and show that x'x = 1 and x'Ax = t. %3D

Let A be an n x n symmetric matrix, let M and m denote the maximum and minimum values of the quadratic form XTAX, and denote corresponding unit eigenvectors by ul and un. The following calculations show that given any numbert between M and m, there is a unit vector x such that t = x"Ax. Verify that t = (1 – a)m + ¤M for some number a between O and 1. Then let x = V1– au, + Jauj, and show that x'x = 1 and x'Ax = t. %3D

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 26EQ

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