8.12Given a matrixA =1В0α0100 1where a and ß are non-zero complex numbers, find its eigenvalues and eigenvec-tors. Find the respective conditions for (a) the eigenvalues to be real and (b) theeigenvectors to be orthogonal. Show that the conditions are jointly satisfied ifand only if A is Hermitian.

To find the eigenvalues and eigenvectors along with conditions.

Answered: A = 1 α B 1 00 0 0 1

Math

Advanced Math

A = 1 α B 1 00 0 0 1

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 11AEXP

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8.12
Given a matrix
A =
1
В
0
α
0
1
0
0 1
where a and ß are non-zero complex numbers, find its eigenvalues and eigenvec-
tors. Find the respective conditions for (a) the eigenvalues to be real and (b) the
eigenvectors to be orthogonal. Show that the conditions are jointly satisfied if
and only if A is Hermitian.

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