(a) Show that if A has the repeated eigenvalue 1 with two linearly independent associated eigenvectors, then every nonzero vector v is an eigenvector of A. (Hint: Express v as a linear combination of the linearly independent eigen- vectors and multiply both sides by A.) (b) Conclude that A must be given by Eq. (22). (Suggestion: In the equation Av = Av take v = [1 0]' and v = [0 1]'.)
(a) Show that if A has the repeated eigenvalue 1 with two linearly independent associated eigenvectors, then every nonzero vector v is an eigenvector of A. (Hint: Express v as a linear combination of the linearly independent eigen- vectors and multiply both sides by A.) (b) Conclude that A must be given by Eq. (22). (Suggestion: In the equation Av = Av take v = [1 0]' and v = [0 1]'.)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.1: Inner Product Spaces
Problem 11AEXP
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Question
Here A represents a 2 x 2 matrix
![(a) Show that if A has the repeated eigenvalue 1 with two
linearly independent associated eigenvectors, then every
nonzero vector v is an eigenvector of A. (Hint: Express v
as a linear combination of the linearly independent eigen-
vectors and multiply both sides by A.) (b) Conclude that
A must be given by Eq. (22). (Suggestion: In the equation
Av = Av take v = [1 0]' and v = [0 1]'.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77199bf4-8d64-4271-9fa2-bb41dab1a2f5%2F7263e55b-dd01-4f15-8317-50ff40677d6f%2Fo0lu55s.png&w=3840&q=75)
Transcribed Image Text:(a) Show that if A has the repeated eigenvalue 1 with two
linearly independent associated eigenvectors, then every
nonzero vector v is an eigenvector of A. (Hint: Express v
as a linear combination of the linearly independent eigen-
vectors and multiply both sides by A.) (b) Conclude that
A must be given by Eq. (22). (Suggestion: In the equation
Av = Av take v = [1 0]' and v = [0 1]'.)
Expert Solution
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Step 1
Given : is a matrix and has the repeated eigen value with two linearly independent eigen vectors.
To prove : Every non zero eigen vector is an eigen vector of .
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