1. Use the following definition of eigenvectors and eigenvalues of a matrix A to prove the fact below: Definition 1: An eigenvector of an n x n matrix A is a nonzero vector such that AA for some scalar A. A scalar A is called an eigenvalue of A if there is a nontrivial solution to A=Xz. Prove: If matrix A, nxn, has two eigenvectors, w₁, 2 corresponding to two distinct eigenvalues A₁, A2 then the set {₁, 2} is linearly independent.

1. Use the following definition of eigenvectors and eigenvalues of a matrix A to prove the fact below: Definition 1: An eigenvector of an n x n matrix A is a nonzero vector such that AA for some scalar A. A scalar A is called an eigenvalue of A if there is a nontrivial solution to A=Xz. Prove: If matrix A, nxn, has two eigenvectors, w₁, 2 corresponding to two distinct eigenvalues A₁, A2 then the set {₁, 2} is linearly independent.

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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