62
3 Eigenvalue Problem
Proof The proof of this statement can be found in, e.g., [1] (Sect. 6.3). This
reference also shows that the eigenvectors and the generalized eigenvectors span
the n-dimensional space (see also Appendix C).
We also note that if {x 1 , . . . , x q } are eigenvectors of an eigenvalue λ 1 , linear
forms of these eigenvectors are also eigenvectors of λ 1 . Let α 1 x 1 + · · · + α q x q
be such a form, where {α j }, j = 1, . . . , q, are arbitrary coefficients. We have
a
α 1 x 1 + · · · + α q x q
= α 1 a x 1 + · · · + α q a x q
= λ 1
α 1 x 1 + · · · + α q x q
so that α 1 x 1 + · · · + α q x q is an eigenvector of λ 1 .
We do not expand further on properties of generalized eigenvectors to keep
the presentation condensed and focused on typical applications. The above comments are intended to (1) recognize that multiple eigenvalues are encountered in
applications and (2) provide a framework for the analysis of dynamical systems
with multiple eigenvalues. These considerations also apply to the generalized
eigenvectors of nonsymmetric matrices which are discussed in the subsequent
section.
Example 3.4 The (2,2)-matrix
a =
1 0
0 1
has the eigenvalues λ 1 = λ 2 = 1. The first eigenvector x 1 is the solution of
a x 1 = λ 1 x 1 or x 1 = x 1 so that the components of this vector are arbitrary
constants constrained by the condition x 1 = 0. The generalized eigenvector x 2 is
defined by a x 2 = λ 1 x 2 +x 1 which gives x 2,1 = x 2,1 +x 1,1 and x 2,2 = x 2,2 +x 1,2 .
If x 1,1 = 0, then x 2,1 = 0 from the first equation. From the second equation, the
solution x 2,1 = 0, and the condition x 2 = 0, we conclude that x 1,2 must be
zero. In summary, the components of x 1 and x 2 are (arbitrary constant, 0) and
(0, arbitrary constant). This result is consistent with our intuition since a is the
identity matrix so that its eigenvectors are aligned with the system of coordinates
in which this matrix is defined.
Example 3.5 The eigenvalues and eigenvectors of the (2,2)-matrix symmetric
matrix
a =
2 1
1 3
delivered by the MATLAB function [u, d] = eig(a) are
3 Eigenvalue Problem
Proof The proof of this statement can be found in, e.g., [1] (Sect. 6.3). This
reference also shows that the eigenvectors and the generalized eigenvectors span
the n-dimensional space (see also Appendix C).
We also note that if {x 1 , . . . , x q } are eigenvectors of an eigenvalue λ 1 , linear
forms of these eigenvectors are also eigenvectors of λ 1 . Let α 1 x 1 + · · · + α q x q
be such a form, where {α j }, j = 1, . . . , q, are arbitrary coefficients. We have
a
α 1 x 1 + · · · + α q x q
= α 1 a x 1 + · · · + α q a x q
= λ 1
α 1 x 1 + · · · + α q x q
so that α 1 x 1 + · · · + α q x q is an eigenvector of λ 1 .
We do not expand further on properties of generalized eigenvectors to keep
the presentation condensed and focused on typical applications. The above comments are intended to (1) recognize that multiple eigenvalues are encountered in
applications and (2) provide a framework for the analysis of dynamical systems
with multiple eigenvalues. These considerations also apply to the generalized
eigenvectors of nonsymmetric matrices which are discussed in the subsequent
section.
Example 3.4 The (2,2)-matrix
a =
1 0
0 1
has the eigenvalues λ 1 = λ 2 = 1. The first eigenvector x 1 is the solution of
a x 1 = λ 1 x 1 or x 1 = x 1 so that the components of this vector are arbitrary
constants constrained by the condition x 1 = 0. The generalized eigenvector x 2 is
defined by a x 2 = λ 1 x 2 +x 1 which gives x 2,1 = x 2,1 +x 1,1 and x 2,2 = x 2,2 +x 1,2 .
If x 1,1 = 0, then x 2,1 = 0 from the first equation. From the second equation, the
solution x 2,1 = 0, and the condition x 2 = 0, we conclude that x 1,2 must be
zero. In summary, the components of x 1 and x 2 are (arbitrary constant, 0) and
(0, arbitrary constant). This result is consistent with our intuition since a is the
identity matrix so that its eigenvectors are aligned with the system of coordinates
in which this matrix is defined.
Example 3.5 The eigenvalues and eigenvectors of the (2,2)-matrix symmetric
matrix
a =
2 1
1 3
delivered by the MATLAB function [u, d] = eig(a) are