Page 146 - Compact Numerical Methods For Computers

Chapter 11

THE GENERALISED SYMMETRIC MATRIX
EIGENVALUE PROBLEM

Consider now the generalised matrix eigenvalue problem
Ax = eBx (2.63)
where A and B are symmetric and B is positive definite. A solution to this
problem will be sought by transforming it into a conventional eigenvalue problem.
The trivial approach
- 1
B Ax = ex (11.1)
- 1
gives an eigenvalue problem of the single matrix B A which is unfortunately not
- 1
symmetric. The approximation to B A generated in a computation may therefore
have complex eigenvalues. Furthermore, methods for solving the eigenproblem of
non-symmetric matrices require much more work than their symmetric matrix
counterparts. Ford and Hall (1974) discuss several transformations that convert
(2.63) into a symmetric matrix eigenproblem.
Since B is positive definite, its eigenvalues can be written as the squares
i = l, 2, . . . , n, so that
2
B=ZD Z T (11.2)
where Z is the matrix of eigenvectors of B normalised so that
T T
ZZ =Z Z= l . (11.3)
n
Then
- 1 T
B -1/2 = ZD Z (11.4)
and
(B -1/2 AB -l/2 )(B 1 / 2 X)=B 1 / 2 XE (11.5)

is equivalent to the complete eigenproblem
AX = BXE (11.5a)
which is simply a matrix form which collects together all solutions to (2.63).
Equation (11.5) can be solved as a conventional symmetric eigenproblem
A V = VE (11.6)
1
where
- 1 / 2 - 1 / 2
A = B AB (11.7a)
1
and
V = B 1 / 2 X. (11.76)
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