Symmetric Matrix Eigenvalue Problems - Numerical Methods in Engineering with MATLAB - page 336

if index ˜= i
eVals = swapRows(eVals,i,index);
eVecs = swapCols(eVecs,i,index);

end

end

Transformation to Standard Form

Physical problems often give rise to eigenvalue problems of the form

Ax = λ Bx (9.21)

×

where A and B aresymmetric n n matrices.We assumethat B is also positive definite.

Such problems must betransformedinto the standard formbefore they can be solved
by Jacobi diagonalization.
As B issymmetric and positive definite, wecan applyCholeski's decomposition

LL T , where L is a lower-triangular matrix (see Art. 2.3). Thenwe introduce the
transformation

B =

( L − 1 ) T z x = (9.22)

Substituting into Eq. (9.21), we get

A ( L − 1 ) T z LL T ( L − 1 ) T z = λ

Premultiplying both sides by L − 1 results in

L − 1 A ( L − 1 ) T z L − 1 LL T ( L − 1 ) T z = λ

Because L − 1 L L T ( L − 1 ) T = = I , the last equationreduces to the standard form

= λ

Hz z (9.23)

where

L − 1 A ( L − 1 ) T

H = (9.24)

An important property of thistransformationisthat it does not destroy the symmetry
of the matrix; i.e., a symmetric A results in a symmetric H .
Here is the general procedurefor solving eigenvalue problems of the form

Ax = λ Bx :

LL T to compute L .
2. Compute L − 1 (a triangular matrix can be invertedwith relatively small computa-
tional effort).
3. Compute H fromEq. (9.24).

1. Use Choleski's decomposition B =