Eigenvectors and Eigenvalues

eigenvals(M) Returns a vector whose elements are the sorted eigenvalues of M.

eigenvec(M, z) Returns a single normalized eigenvector associated with eigenvalue z of M. The eigenvector is normalized to unit length. Uses an inverse iteration algorithm.

eigenvecs(M, ["L"]) Returns a matrix containing all normalized eigenvectors of the matrix M. The nth column of the matrix returned is an eigenvector corresponding to the nth eigenvalue returned by eigenvals. The right eigenvector is returned by default. Can also return the left eigenvector, satisfying v^H·M =z·v^H, where H indicates conjugate transpose.

genvals(M, N) Returns a vector of computed eigenvalues, v_i, each of which satisfies the generalized eigenvalue problem M·x = v_i·N·x for its associated eigenvector x_i.

genvecs(M, N, ["L"]) Returns a matrix containing the normalized eigenvectors corresponding to the eigenvalues in v, the vector returned by genvals. The ith column of this matrix is the eigenvector x satisfying the generalized eigenvalue problem.

tr(M) Returns the trace of M, that is, the sum of the elements along the diagonal of M. This is equal to the sum of the eigenvalues.

The BLAS/LAPACK libraries (http://www.intel.com/software/products/mkl/features/lin_alg.htm) from Intel are used in all these functions.

Arguments:

• M, N are square matrices of real or complex numbers of equal size.
• "L" (optional) is a string. When used, the string "L" specifies the left eigenvector, and "R" the right. "R" is the default.
• z is an eigenvalue of M.

Notes:

• The results returned by eigenvec and eigenvecs are not necessarily identical. For a given eigenvalue, there are infinitely many eigenvectors, and the one found depends on the algorithm used. Every eigenvector for a particular eigenvalue is a multiple of other eigenvectors.
• The symbolic processor doesn't necessarily produce identical results with numerical evaluation, for the same reasons. There may be different normalizations, sign ambiguities, and other differences in the two types of results, although all are correct. Eigenvectors are returned in largest to smallest order for both results. The eigenvec function is not available symbolically, but eigenvecs is.
• You may wish to solve linear systems of equations using lsolve, or a solve block.
• You may wish to check whether your matrix is singular, or nearly so, using the condition number.

QuickSheet