Symmetric Matrix Eigenvalue Problems - Numerical Methods in Engineering with MATLAB - page 328

Here aresome useful properties of eigenvalues and eigenvectors, givenwithout
proof:

All eigenvalues of a symmetric matrix are real.
All eigenvalues of a symmetric, positive-definite matrix are real and positive.

The eigenvectorsofa symmetric matrix areorthonormal; that is, X T X = I .

λ i , then the eigenvalues of A − 1 are λ − 1

i

If the eigenvalues of A are .

Eigenvalue problemsthatoriginate fromphysical problems often end up with
a symmetric A . This isfortunate, because symmetriceigenvalue problems are much
easier to solvethantheirnonsymmetriccounterparts. In thischapterwelargelyrestrict
our discussion to eigenvalues and eigenvectorsofsymmetric matrices.
Common sources of eigenvalue problems are the analysisofvibrations and sta-
bility. These problems often have the following characteristics:

The matrices arelarge and sparse (e.g.,have a banded structure).
We need to know only the eigenvalues;ifeigenvectors are required,onlyafew of
themareofinterest.

Ausefuleigenvalue solvermust be able to utilize these characteristics tominimize
the computations. In particular, it shouldbe flexible enoughtocompute onlywhat
we need and no more.

9.2 Jacobi Method

Similarity Transformation and Diagonalization

Consider the standard matrix eigenvalue problem

Ax = λ x (9.4)

where A issymmetric.Let us now apply the transformation

Px ∗ x = (9.5)

where P is anonsingularmatrix. SubstitutingEq. (9.5) into Eq. (9.4) andpremultiplying
each side by P − 1 , we get

P − 1 APx ∗ = λ P − 1 Px ∗

or

A ∗ x ∗ = λ x ∗

(9.6)