Let the matrix A be given. Suppose that v is an eigenvector of A with corresponding eigenvalue λ, so Prove that Αν = λν. exp(A)v=ev Use this result to show the eigenspace of A is invariant under the differential equa- tion d dt = x(t) = Ax(t). Invariant here means: if x(0) = μv for some scalar µ and eigenvector v, then x(t) = cv for all time. Hint: Consider the definition of exp(A) applied to v.

Let the matrix A be given. Suppose that v is an eigenvector of A with corresponding eigenvalue λ, so Prove that Αν = λν. exp(A)v=ev Use this result to show the eigenspace of A is invariant under the differential equa- tion d dt = x(t) = Ax(t). Invariant here means: if x(0) = μv for some scalar µ and eigenvector v, then x(t) = cv for all time. Hint: Consider the definition of exp(A) applied to v.

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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