Generalized Eigenvectors - University of Pennsylvania
Generalized Eigenvectors
Math 240
Definition Computation and Properties Chains
Generalized Eigenvectors
Math 240 -- Calculus III
Summer 2013, Session II
Wednesday, July 31, 2013
Generalized Eigenvectors
Math 240
Definition Computation and Properties Chains
1. Definition
2. Computation and Properties
3. Chains
Agenda
Generalized Eigenvectors
Math 240
Definition Computation and Properties Chains
Motivation
Defective matrices cannot be diagonalized because they do not possess enough eigenvectors to make a basis. How can we correct this defect?
Example
The matrix A =
1 0
1 1
is defective.
1. Only eigenvalue is = 1.
2. A - I =
0 0
1 0
3. Single eigenvector v = (1, 0).
4. We could use u = (0, 1) to complete a basis.
5. Notice that (A - I)u = v and (A - I)2u = 0.
Maybe we just didn't multiply by A - I enough times.
Generalized Eigenvectors
Math 240
Definition Computation and Properties Chains
Definition
Definition
If A is an n ? n matrix, a generalized eigenvector of A
corresponding to the eigenvalue is a nonzero vector x
satisfying
(A - I)p x = 0
for some positive integer p. Equivalently, it is a nonzero element of the nullspace of (A - I)p.
Example
Eigenvectors are generalized eigenvectors with p = 1.
In the previous example we saw that v = (1, 0) and u = (0, 1) are generalized eigenvectors for
A=
1 0
1 1
and = 1.
Generalized Eigenvectors
Math 240
Definition Computation and Properties Chains
Computing generalized eigenvectors
Example
Determine generalized eigenvectors for the matrix 1 1 0
A = 0 1 2 . 003
1. Characteristic polynomial is (3 - )(1 - )2.
2. Eigenvalues are = 1, 3.
3. Eigenvectors are
1 = 3 :
v1 = (1, 2, 2),
2 = 1 :
v2 = (1, 0, 0).
4. Final generalized eigenvector will a vector v3 = 0 such that
(A - 2I)2 v3 = 0 but (A - 2I) v3 = 0.
Pick v3 = (0, 1, 0). Note that (A - 2I)v3 = v2.
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