Numerical recipes

462 Chapter 11. Eigensystems
§11.6)
We also discuss, in §11.7, how to obtain some eigenvectors of nonsymmetric
matrices by the method of inverse iteration.
Generalized and Nonlinear Eigenvalue Problems
Many eigenpackages also deal with the so-called generalized eigenproblem, [6]
A · x = λB · x (11.0.18)
where A and B are both matrices. Most such problems, where B is nonsingular,
can be handled by the equivalent
(B −1 · A) · x = λx (11.0.19)
Often A and B are symmetric and B is positive definite. The matrix B −1 · A in
(11.0.19) is not symmetric, but we can recover a symmetric eigenvalue problem
by using the Cholesky decomposition B = L · L T of §2.9. Multiplying equation
(11.0.18) by L −1 , we get
where
C · (L T · x) =λ(L T · x) (11.0.20)
C = L −1 · A · (L −1 ) T
(11.0.21)
The matrix C is symmetric and its eigenvalues are the same as those of the original
problem (11.0.18); its eigenfunctions are L T · x. The efficient way to form C is
first to solve the equation
for the lower triangle of the matrix Y. Then solve
Y · L T = A (11.0.22)
L · C = Y (11.0.23)
for the lower triangle of the symmetric matrix C.
Another generalization of the standard eigenvalue problem is to problems
nonlinear in the eigenvalue λ, for example,
(Aλ 2 + Bλ + C) · x =0 (11.0.24)
This can be turned into a linear problem by introducing an additional unknown
eigenvector y and solving the 2N × 2N eigensystem,
�
0 1
−A −1 · C −A −1 · B
�
·
� �
x
= λ
y
� �
x
y
(11.0.25)
This technique generalizes to higher-order polynomials in λ. A polynomial of degree
M produces a linear MN × MN eigensystem (see [7]).
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