TABLE OF CONTENTS This chapter is about eigenvalues and singular values of matrices. Computational algorithms and sensitivity to perturbations are both discussed.
An eigenvalue and eigenvector of a square matrix A are a scalar ? and a nonzero vector x so that
A singular value and pair of singular vectors of a square or rectangular matrix A are a nonnegative scalar ? and two nonzero vectors u and v so that
The superscript on A H stands for Hermitian transpose and denotes the complex conjugate transpose of a complex matrix. If the matrix is real, then A T denotes the same matrix. In MATLAB, these transposed matrices are denoted by A ?.
The term "eigenvalue" is a partial translation of the German "eigenvert." A complete translation would be something like "own value" or "characteristic value," but these are rarely used. The term "singular value" relates to the distance between a matrix and the set of singular matrices.
Eigenvalues play an important role in situations where the matrix is a transformation from one vector space onto itself. System of linear ordinary differential equations are the primary examples. The values of ? can correspond to frequencies of vibration, or critical values of stability parameters, or energy levels of atoms. Singular values play an important role where the matrix is a transformation from one vector space to a different vector space, possibly with a...
