The **nonsymmetric eigenvalue problem** is to find the **eigenvalues**, ,
and corresponding **eigenvectors**, , such that

A real matrix may have complex eigenvalues, occurring as complex conjugate pairs. More precisely, the vector is called a

is called a

This problem can be solved via the

where is an orthogonal matrix and is an upper quasi-triangular matrix with and diagonal blocks, the blocks corresponding to complex conjugate pairs of eigenvalues of . In the complex case the Schur factorization is

where is unitary and is a complex upper triangular matrix.

The columns of are called the

Two pairs of drivers are provided, one pair focusing on the Schur factorization, and the other pair on the eigenvalues and eigenvectors as shown in Table 2.5:

- LA_GEES: a simple driver that computes all or
part of the Schur factorization of , with optional ordering of the
eigenvalues;

- LA_GEESX: an expert driver that can additionally
compute condition numbers for the average of a selected subset of the
eigenvalues, and for the corresponding right invariant subspace;

- LA_GEEV: a simple driver that computes all the
eigenvalues of , and (optionally) the right or left eigenvectors (or both);

- LA_GEEVX: an expert driver that can additionally balance the matrix to improve the conditioning of the eigenvalues and eigenvectors, and compute condition numbers for the eigenvalues or right eigenvectors (or both).