3 3. Let A be an n x n matrix. Let (x, y) = x'y be the scalar product for R".(a) Show that (Ax, y) = (x, ATy) for all x, y € R".(b) Suppose that A is symmetric, i.e., A" = A. Suppose that x and y are eigenvectorssatisfying Axshow that x ly.Ax and Ay = µy with A, µ E R and A # µ. Show that (x, y) = 0, i.e.,

Given that A is a n×n matrix. (a) We have to show Ax,y=x,ATy ∀ x,y∈ℝn.

3. Let A be an n x n matrix. Let (x, y) = x"y be the scalar product for R". (a) Show that (Ax, y) = (x, A" y) for all x, y € R". (b) Suppose that A is symmetric, i.e., AT = A. Suppose that x and y are eigenvectors satisfying Ax = \x and Ay = µy with A, µ e R and A # µ. Show that (x, y) = 0, i.e., show that x I y.

3. Let A be an n x n matrix. Let (x, y) = x"y be the scalar product for R". (a) Show that (Ax, y) = (x, A" y) for all x, y € R". (b) Suppose that A is symmetric, i.e., AT = A. Suppose that x and y are eigenvectors satisfying Ax = \x and Ay = µy with A, µ e R and A # µ. Show that (x, y) = 0, i.e., show that x I y.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 27EQ

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