# lapgeigv¶

## Purpose¶

Computes generalized eigenvalues, left eigenvectors, and right eigenvectors for a pair of real or complex general matrices.

## Format¶

{ va1, va2, lve, rve } = lapgeigv(A, B)
Parameters
• A (NxN matrix) – real or complex general matrix.

• B (NxN matrix) – real or complex general matrix.

Returns
• va1 (Nx1 vector) – numerator of eigenvalues.

• va2 (Nx1 vector) – denominator of eigenvalues.

• lve (NxN left eigenvectors) –

• rve (NxN right eigenvectors) –

## Remarks¶

va1 and va2 are the vectors of the numerators and denominators respectively of the eigenvalues of the solution of the generalized symmetric eigenproblem of the form $$Aw = \lambda Bw$$ where A and B are real or complex general matrices and $$w = va1./va2$$. The generalized eigenvalues are not computed directly because some elements of va2 may be zero, i.e., the eigenvalues may be infinite.

The left and right eigenvectors diagonalize $$U'^{-1}AU^{-1}$$ where $$B = U'U$$, that is,

$\text{lve}*U'^{-1}AU^{-1}*\text{lve}' = w$

and

$\text{rve}'*U'^{-1}AU^{-1}*\text{rve} = w$

This procedure calls the LAPACK routines DGGEV and ZGGEV.