First, give the definition of exp(A) for a given matrix A. Next, write down what it means for v; to be an eigenvector with eigenvalue λ of the matrix A. Then, determine the eigenvalues and eigenvectors of M = exp(A). Use this result to prove that M is invertible. Finally, recall that if two matrices U, V commute (UV = VU), then exp(U+V) = exp(U) exp(V). Use this result to find the inverse of M.

First, give the definition of exp(A) for a given matrix A. Next, write down what it means for v; to be an eigenvector with eigenvalue λ of the matrix A. Then, determine the eigenvalues and eigenvectors of M = exp(A). Use this result to prove that M is invertible. Finally, recall that if two matrices U, V commute (UV = VU), then exp(U+V) = exp(U) exp(V). Use this result to find the inverse of M.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 26EQ

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