(i) Let A be an invertible nxn matrix. Given the eigenvalues and eigenvectors of A. What can be said about the eigenvalues of A®A-1+A-l®A? (ii) Let c e R and A be a normal n x n matrix. Assume that A is invertible. The eigenvalue equation is given by Av; = 1,v; for j = 1,...,n and A; #0 for all j = 1,...,n. Find all the eigenvalues and eigenvectors of the matrix A® A-1 +c(A¬1 ® A). Find det(A ® A-1) and tr(A ® A-1).

(i) Let A be an invertible nxn matrix. Given the eigenvalues and eigenvectors of A. What can be said about the eigenvalues of A®A-1+A-l®A? (ii) Let c e R and A be a normal n x n matrix. Assume that A is invertible. The eigenvalue equation is given by Av; = 1,v; for j = 1,...,n and A; #0 for all j = 1,...,n. Find all the eigenvalues and eigenvectors of the matrix A® A-1 +c(A¬1 ® A). Find det(A ® A-1) and tr(A ® A-1).

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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