Given two m ×m matrix X and Y , where XY = Y X. 1) Let u be an eigenvector of X. Show that either Y u is an eigenvector of X or Y u is a zero vector. 2) Suppose Y is invertible and Y u is an eigenvector of X. Show u is an eigen- vector of X.

Given two m ×m matrix X and Y , where XY = Y X. 1) Let u be an eigenvector of X. Show that either Y u is an eigenvector of X or Y u is a zero vector. 2) Suppose Y is invertible and Y u is an eigenvector of X. Show u is an eigen- vector of X.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 23EQ

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Question

Given two m ×m matrix X and Y , where XY = Y X.
1) Let u be an eigenvector of X. Show that either Y u is an eigenvector of X or
Y u is a zero vector.
2) Suppose Y is invertible and Y u is an eigenvector of X. Show u is an eigen-
vector of X.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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