lapgeighv#

Purpose#

Computes generalized eigenvalues and eigenvectors for a pair of real symmetric or Hermitian matrices.

Format#

{ ve, va } = lapgeighv(A, B)#

Parameters:

A (NxN matrix) – real or complex symmetric or Hermitian matrix.
B (NxN matrix) – real or complex positive definite symmetric or Hermitian matrix.

Returns:

ve (Nx1 vector) – eigenvalues.
va (NxN matrix) – eigenvectors.

Examples#

// Assign A
A = { 3 4 5,
      2 5 2,
      3 2 4 };

// Assign B
B = { 4 2 2,
      2 6 1,
      2 1 8 };

// Find the eigenvalues and corresponding
// eigenvectors of the solution of the
// generalized symmetric eigenproblem
{ ve, va } = lapgeighv(A, B);

print ve;

-0.0425
 0.5082
 0.8694

print va;

 0.3575 -0.0996 0.9286
-0.2594  0.9446 0.2012
-0.8972 -0.3128 0.3118

Remarks#

ve and va are the eigenvalues and eigenvectors of the solution of the generalized symmetric eigenproblem of the form \(Ax = λ B\). Equivalently, va diagonalizes \(U'^{-1}AU^{-1}\) in the following way

\[va*U'^{-1}AU^{-1}*va' = ve\]

where \(B = U'U\). This procedure calls the LAPACK routines DSYGV and ZHEGV.