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
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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