EigenDecomp

EigenDecomp( A, I, J )

Computes the Eigenvalues and Eigenvectors of a square symmetric matrix A indexed by I and J. Eigenvalues and Eigenvectors are also called characteristic values and characteristic vectors.

The pairs of Eigenvalues and Eigenvectors returned are indexed by J. If B is the result of evaluating EigenDecomp(A,I,J), then the Eigenvalues are given by

B[.item='value']

and the Eigenvectors are given by

#B[.item='vector']

Each Eigenvector is indexed by I.

Library

Matrix

Testing for Definiteness

A square matrix A is positive definite if for for every non-zero vector x, the matrix product x'Ax > 0. The matrix A is positive definite if and only if all eigenvalues are positive, and thus the following expression tests for positive-definiteness:

 Function IsPosDefinite(A:Array[I,J] ; I,J : Index) :=
   min(EigenDecomp(A,I,J)[.item='value']>0,J)

Square matrix A is positive semi-definite when x'Ax >= 0 for every vector x, which implies that the eigenvectors are all non-negative.

 Function IsPosSemiDefinite(A:Array[I,J] ; I,J : Index) :=
   min(EigenDecomp(A,I,J)[.item='value']>=0,J)
 

Negative definite and Negative-semidefinite are tested for similarly:

 Function IsNegDefinite(A:Array[I,J] ; I,J : Index) :=
   min(EigenDecomp(A,I,J)[.item='value']<0,J)

 Function IsNegSemiDefinite(A:Array[I,J] ; I,J : Index) :=
   min(EigenDecomp(A,I,J)[.item='value']<=0,J)

Principle Components

(to fill in)