Suppose that the matrix A has repeated eigenvalue with the following eigenvector and generalized eigenvector: A = -2 with eigenvector v = H and generalized eigenvector w = [] Write the solution to the linear system 7' = Ar in the following forms. In eigenvalue/eigenvector form: [3(6)] = C1 help (matrices) [8] In fundamental matrix form: [3 (4)] = + C2 e 381 a help (formulas) help (matrices) [8] + - [B] As two equations: (write "c1" and "c2" for C1 and C2 ) x(t) y(t) = = help (formulas) help (formulas) e Note: If you are feeling adventurous you could use other eigenvectors like 4 v and other generalized eigenvectors like w - 3v. Just remember that if you change v, you must also change w for its fundamental solution! Book: Section 3.7 of Notes on Diffy Qs

Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.1: Eigenvalues And Eigenvectors
Problem 66E: Show that A=[0110] has no real eigenvalues.
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Suppose that the matrix A has repeated eigenvalue with the following eigenvector and
generalized eigenvector:
A = -2 with eigenvector v
=
H
and generalized eigenvector w
=
[]
Write the solution to the linear system 7' = Ar in the following forms.
In eigenvalue/eigenvector form:
[3(6)]
= C1
help (matrices)
[8]
In fundamental matrix form:
[3 (4)]
=
+ C2
e
381 a
help (formulas) help (matrices)
[8] + - [B]
As two equations: (write "c1" and "c2" for C1 and C2 )
x(t)
y(t)
=
=
help (formulas)
help (formulas)
e
Note: If you are feeling adventurous you could use other eigenvectors like 4 v and other
generalized eigenvectors like w - 3v.
Just remember that if you change v, you must also change w for its fundamental solution!
Book: Section 3.7 of Notes on Diffy Qs
Transcribed Image Text:Suppose that the matrix A has repeated eigenvalue with the following eigenvector and generalized eigenvector: A = -2 with eigenvector v = H and generalized eigenvector w = [] Write the solution to the linear system 7' = Ar in the following forms. In eigenvalue/eigenvector form: [3(6)] = C1 help (matrices) [8] In fundamental matrix form: [3 (4)] = + C2 e 381 a help (formulas) help (matrices) [8] + - [B] As two equations: (write "c1" and "c2" for C1 and C2 ) x(t) y(t) = = help (formulas) help (formulas) e Note: If you are feeling adventurous you could use other eigenvectors like 4 v and other generalized eigenvectors like w - 3v. Just remember that if you change v, you must also change w for its fundamental solution! Book: Section 3.7 of Notes on Diffy Qs
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