Eigenvalues and Singular Values 10.1 Eigenvalue and Singular Value Decompositions

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 λ and a nonzero vector x so that Ax = λx. A singular value and pair of singular vectors of a square or rectangular matrix A are a nonnegative scalar σ and two nonzero vectors u and v so that Av = σu, A H u = σv. 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. Systems of linear ordinary differential equations are the primary examples. The values of λ 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 different dimension. Systems of over-or underdetermined algebraic equations are the primary examples.