Just do part III & iv A =Го110010 1 0Consider an Nx N matrix A with N orthonormal eigenvectors x such that Ax' = x¹,where the X, is the eigenvalue corresponding to eigenvector x'. It can be shown that sucha matrix A has an expansion of the form:A=[A₁|x¹)(x¹=[A₁x²(x¹)¹.i) Show that if the eigenvalues are real then A, as defined through the above expansion,is Hermitian.ii) Using the result for A show that the Nx N identity matrix can be written as1=[x²(x¹)¹.iii) In proving this result for the identity matrix you have used the fact that the vectors{x} are eigenvectors of a matrix A. Is this essential or is there a milder requirementpossible? If so, what is it? Justify your answers.iv) Verify the above identity for I using the eigenvectors of A from the first part of thisIf you have been unable to find suitable eigenvectors of the A, then use the followingset for this part of the question: x¹ = (0, 0, 1)¹, x² = (1/√2) (1, 1, 0)T, and x³ =(1/√2) (1, 1, 0)T. Note, these are not the eigenvectors of A.

Answered: Image /qna-images/answer/f0ec34fc-e05a-48ca-bb47-271e784e16e6.jpg

A = 1 1 10 Го 0 1 0 1 0 Consider an Nx N matrix A with N orthonormal eigenvectors x such that Ax' = A₁x¹, where the X, is the eigenvalue corresponding to eigenvector x'. It can be shown that such a matrix A has an expansion of the form: A=A₁|x¹)(x¹=EA;x²(x¹). i) Show that if the eigenvalues are real then A, as defined through the above expansion, is Hermitian. ii) Using the result for A show that the Nx N identity matrix can be written as

A = 1 1 10 Го 0 1 0 1 0 Consider an Nx N matrix A with N orthonormal eigenvectors x such that Ax' = A₁x¹, where the X, is the eigenvalue corresponding to eigenvector x'. It can be shown that such a matrix A has an expansion of the form: A=A₁|x¹)(x¹=EA;x²(x¹). i) Show that if the eigenvalues are real then A, as defined through the above expansion, is Hermitian. ii) Using the result for A show that the Nx N identity matrix can be written as

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.1: Introduction To Eigenvalues And Eigenvectors

Problem 35EQ

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