Let A and B be n-by-n matrices. A scalar is called a generalized eigenvalue and a non-zero column vector x the corresponding right generalized eigenvector of the pair (A,B), if . A non-zero column vector y satisfying is called the left generalized eigenvector corresponding to . (For simplicity, we will usually omit the word ``generalized'' when no confusion is likely to arise.) If B is singular, we can have the infinite eigenvalue , by which we mean Bx = 0. Note that if A is non-singular, then the equivalent problem is perfectly well-defined, and the infinite eigenvalue corresponds to . The generalized symmetric definite eigenproblem in section 2.3.7 has only finite real eigenvalues. The generalized nonsymmetric eigenvalue problem can have real, complex or infinite eigenvalues. To deal with both finite (including zero) and infinite eigenvalues, the LAPACK routines return two values, and . If is non-zero then is an eigenvalue. If is zero then is an eigenvalue of (A, B). (Round off may change an exactly zero to a small nonzero value, changing the eigenvalue to some very large value; see section 4.11 for details.) A basic task of these routines is to compute all n pairs and x and/or y for a given pair of matrices (A,B).
If the determinant of is identically zero for all values of , the eigenvalue problem is called singular; otherwise it is regular. Singularity of (A,B) is signaled by some (in the presence of roundoff, and may be very small). In this case, the eigenvalue problem is very ill-conditioned, and in fact some of the other nonzero values of and may be indeterminate (see section 18.104.22.168 for further discussion) [93,105,29,53].
Another basic task is to compute the generalized Schur decomposition of the pair (A,B). If A and B are complex, then their generalized Schur decomposition is A = QSZH and B = QTZH, where Q and Z are unitary and S and T are upper triangular. The LAPACK routines normalize T to have real non-negative diagonal entries. Note that in this form, the eigenvalues can be easily computed from the diagonals: (if ) and (if tii = 0), and so the LAPACK routines return and .
The generalized Schur form depends on the order of the eigenvalues on the diagonal of (S,T). This order may optionally be chosen by the user.
If A and B are real, then their generalized Schur decomposition
is A = QSZT and B = QTZT, where Q and Z are orthogonal,
S is quasi-upper triangular with 1-by-1 and 2-by-2 blocks on the
diagonal, and T is upper triangular with non-negative diagonal entries.
The structure of a typical pair of (S,T) is illustrated below for n=6:
The columns of Q and Z are called generalized Schur vectors
and span pairs of deflating subspaces of A and B .
Deflating subspaces are a generalization of invariant subspaces: the first k
columns of Z span a right deflating subspace mapped by both A and
B into a left deflating subspace spanned by the first k columns of
Q. This pair of deflating subspaces corresponds to the first k
eigenvalues appearing at the top left corner of S and T as explained
in section 22.214.171.124.
The computations proceed in the following stages:
Other subsidiary tasks may be performed before or after those described.