Page 38 - Compact Numerical Methods For Computers

28 Compact numerical methods for computers
+ + + T T + T
A AA = VS U USV VS U
T
+
+
+
T
= VS SS U = VS U = A + (2.60)
and
T T
+
T
+
+
T T
T
(A A) = (VS U USV ) = (VS SV )
+ T +
= VS SV = A A. (2.61)
Several of the above relationships depend on the diagonal nature of S and S + and
on the fact that diagonal matrices commute under multiplication.
2.6. THE MATRIX EIGENVALUE PROBLEM
An eigenvalue e and eigenvector x of an n by n matrix A, real or complex, are
respectively a scalar and vector which together satisfy the equation
Ax = ex. (2.62)
There will be up to n eigensolutions (e, x) for any matrix (Wilkinson 1965) and
finding them for various types of matrices has given rise to a rich literature. In
many cases, solutions to the generalised eigenproblem

Ax = eBx (2.63)
are wanted, where B is another n by n matrix. For matrices which are of a size
that the computer can accommodate, it is usual to transform (2.63) into type
(2.62) if this is possible. For large matrices, an attempt is usually made to solve
(2.63) itself for one or more eigensolutions. In all the cases where the author has
encountered equation (2.63) with large matrices, A and B have fortunately been
symmetric, which provides several convenient simplifications, both theoretical and
computational.
Example 2.5. Illustration of the matrix eigenvalue problem

In quantum mechanics, the use of the variation method to determine approximate
energy states of physical systems gives rise to matrix eigenvalue problems if the
trial functions used are linear combinations of some basis functions (see, for
instance, Pauling and Wilson 1935, p 180ff).
If the trial function is F, and the energy of the physical system in question is
described by the Hamiltonian operator H, then the variation principle seeks
stationary values of the energy functional
(F, HF)
C =
(E, F) (2.64)

subject to the normalisation condition

(F, F) = 1 (2.65)
where the symbol ( , ) represents an inner product between the elements
separated by the comma within the parentheses. This is usually an integral over all