## Generalized Eigenvalues, Schur Form, and Schur Vectors of General Matrices

The subroutines in this section compute the generalized eigenvalues (, ) for a pair of square general matrices A and B, and compute the Schur form
(A,B) = (VLTAVR, VLTBVR) for real matrices or the Schur form
(A,B) = (VLHAVR, VLHBVR) for complex matrices where VL and VR are the left and right Schur vectors. These subroutines can optionally return the left and/or right Schur vectors. Note that the driver xGEGV is also available.

A generalized eigenvalue is a ratio = / such that AX = BX or
det(A-B) = 0 for general matrices A, B, and X. is typically represented as a ratio rather than as a scalar because there are reasonable interpretations for = 0, = 0, and for = = 0. A right generalized eigenvector x corresponding to a generalized eigenvalue is defined by (A - B)x = 0. A left generalized eigenvector x corresponding to a generalized eigenvalue is defined by
(A - B)Hx = 0. A good reference for generalized eigenproblems is the book Matrix Computations, 2nd. ed. by Golub and van Loan (1989, The Johns Hopkins University Press).