Nonsymmetric Eigenproblems (NEP)

The nonsymmetric eigenvalue problem is to find the eigenvalues, $\lambda$, and corresponding eigenvectors, $v \ne 0$, such that

$$Av = \lambda v.$$

A real matrix A may have complex eigenvalues, occurring as complex conjugate pairs. More precisely, the vector v is called a right eigenvector of A, and a vector $u\neq 0$ satisfying

$$u^HA = \lambda u^H$$

is called a left eigenvector of A.

This problem can be solved via the Schur factorization of A, defined in the real case as

A = ZTZ^T,

where Z is an orthogonal matrix and T 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. In the complex case the Schur factorization is

A = ZTZ^H,

where Z is unitary and T is a complex upper triangular matrix.

The columns of Z are called the Schur vectors. For each k $(1 \leq k \leq n)$, the first k columns of Z form an orthonormal basis for the invariant subspace corresponding to the first k eigenvalues on the diagonal of T. Because this basis is orthonormal, it is preferable in many applications to compute Schur vectors rather than eigenvectors. It is possible to order the Schur factorization so that any desired set of k eigenvalues occupy the k leading positions on the diagonal of T.

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:

• xGEES: a simple driver that computes all or part of the Schur factorization of A, with optional ordering of the eigenvalues;

• xGEESX: 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;

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

• xGEEVX: 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).