conjorth

Constructs conjugate-orthogonal basis from matrix

Compute a set of normalized, A-conjugate vectors from basis W. Vectors can be A-normalized outside of function easily. See below.

[U]=conjorth(W,A)

INPUT
W: A linear independet set of column vectors

A: A full rank matrix. An output is produced if A is not symmetric, but A must be symmetric in order for U'*A*U=0;

OUTPUT
U: A set of mutually A-conjuate vectors that span the same space as W. If A if the identity matrix, so that A=eye(size(W)), U is a set of orthogonal vectors and the method is identical to the Gram-Schmidt
orthogonalization. The columns of U are normalized.

To A-normalize U(:,k), compute:
U(:,k)/(U(:,k)'*A*U(:,k))

DESCRIPTION OF A-CONJUGACY
Two vectors u and v are A-conjugate if u'*(A*v) = 0. In a more general sense, two elements of a vector space are A-conjugate if <u,Av>=0. A-conjugacy arises out of generalized eigenvalue problems of the form:

Lu = lambda*A*u

If A is self-adjoint and positive definite, and L is self-adjoint, then the eigenvectors of operator L are A-conjugate to each other. Such problems arise all the time in vibrational mechanics. The Ritz method is a common aplication of A-conjugacy of eigenvectors. In seismology, non-spherical bodies will in general, have their wave-equation
eigenvectors A-conjugate to each other, rather than orthogonal, though the elements of A are likely to be quite small for the Earth.

USE WITH:
Rayleigh-Ritz algorithms, polarization algorithms, Gram-Schmidt orthogonalization, eigenvalue problems.