Eigenvalues and -vectors

Symmetric Eigenproblems

Eigenvalues λ_i of a square or hermitian matrix A are those scalar values for which vectors v_i≠0 exist such that

In both cases the Eigenvalues λ are real. Considering the sets of all Eigenvalues λ₁,...,λ_k and Eigenvectors v₁,...,v_k of a matrix A we may write

where Λ is the diagonal matrix whose diagonal elements are the Eigenvalues and V is the orthogonal (or unitary) matrix whose columns are the corresponding Eigenvectors.

Non-Symmetric Eigenproblems

The case of non-symmetric matrices A is largely identical to the symmetric case, except that the Eigenvalues λ may be complex. Moreover, in the symmetric case there is no difference between right Eigenvectors (see above) and left Eigenvecors given by

Non square matrices

Eigenvectors and Eigenvalues are only defined for square matrices. For general m x n matrices a related concept exists, the so called singular value decomposition (SVD). For more details please check the next section or the class reference for ILMath.svd.

Generalizes Eigenproblems

Another extension of Eigenproblems are the so called generalized Eigenvalues and Eigenvectors. These are given by the solutions to

where A and B are symmetric (or hermitian) matrices and B is positive definite. This again may be generalized to the non-symmetric case or general m x n matrices (i.e. generalized SVD).

Further Readings

For more details on Eigenvalues, Eigenvectors, and related concepts see e.g. the correponding Wikipedia article. Furthermore, you might want to consult the original LAPACK Documentation providing details about the calculations.

Finding Eigenvectors and Eigenvalues in ILNumerics^®

ILNumerics currently supports the solution of square symmetric and non-symmetric Eigenproblems (SVD as generalization to the non square case is support as well), and the solution of generalized Eigenproblems. Please see the API documentation for further details: