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Matrix.EigGen Method (TMtx, TVec, TVec, TBalanceType, TEigBalancing, TVec, TVec, TMtx, TMTx)
Matrix Record | Matrix Members | MtxExpr Namespace | EigGen Method
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Computes generalized eigenvalues and eigenvectors of a non-symmetric matrix.

Pascal
procedure EigGen(B: TMtx; DAlpha: TVec; DBeta: TVec; Balance: TBalanceType; BInfo: TEigBalancing; rconde: TVec = nil; rcondv: TVec = nil; VL: TMtx = nil; VR: TMTx = nil); overload;
Description

Computes for a pair of N-by-N real nonsymmetric matrices (A = Self,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors (VL and/or VR). 

A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta := lambda, such that A - lambda*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta = 0, and even for both being zero. 

The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies: 

 

A * v(j) = lambda(j) * B * v(j).

 

The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies: 

 

u(j)**H * A = lambda(j) * u(j)**H * B .

 

where u(j)**H is the conjugate-transpose of u(j). The individual eigevalues can be computed as: 

 

lambda(j) = dAlpha(j)/dBeta(j);

 

Optionally also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors , reciprocal condition numbers for the eigenvalues (rconde), and reciprocal condition numbers for the right eigenvectors (rcondv).

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