Let A be a diagonalizable n × n matrix, and let p(λ) = det(A − λI) be the characteristic polynomial of A. Let P be an invertible matrix such that P −1AP = D, where D is the diagonal matrix whose diagonal entries are the eigenvalues of A. Show that trace(A) = trace(D). Then show that the trace of an n × n diagonalizable matrix is the sum of the eigenvalues of that matrix.

Let A be a diagonalizable n × n matrix, and let p(λ) = det(A − λI) be the characteristic polynomial of A. Let P be an invertible matrix such that P −1AP = D, where D is the diagonal matrix whose diagonal entries are the eigenvalues of A. Show that trace(A) = trace(D). Then show that the trace of an n × n diagonalizable matrix is the sum of the eigenvalues of that matrix.

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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Question

Let A be a diagonalizable n × n matrix, and let p(λ) = det(A − λI) be the
characteristic polynomial of A. Let P be an invertible matrix such that P −1AP = D, where
D is the diagonal matrix whose diagonal entries are the eigenvalues of A. Show that trace(A) = trace(D). Then show that the trace of an n × n
diagonalizable matrix is the sum of the eigenvalues of that matrix.

Expert Solution