Given two square matrices A and B, the generalized nonsymmetric eigenvalue problem is to find the eigenvalues and corresponding eigenvectors such that
or find the eigenvalues and corresponding eigenvectors such that
Note that these problems are equivalent with and if neither nor is zero. In order to deal with the case that or is zero, or nearly so, the LAPACK routines return two values, and , for each eigenvalue, such that and .
More precisely, and are called right eigenvectors. Vectors or satisfying
are called left eigenvectors.
If the determinant of is zero for all values of , the eigenvalue problem is called singular, and 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 .
The generalized nonsymmetric eigenvalue problem can be solved via the generalized Schur factorization of the pair A,B, defined in the real case as
where Q and Z are orthogonal matrices, P is upper triangular, and S is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal blocks, the 2-by-2 blocks corresponding to complex conjugate pairs of eigenvalues of A,B. In the complex case the Schur factorization is
where Q and Z are unitary and S and P are both upper triangular.
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: For each k (1 < = k < = n), 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.
Two simple drivers are provided for the nonsymmetric problem :