Matrix Methods

Abstract

Solving systems of linear equations, linear least-square problems and matrix eigenvalue problems is handled by a myriad of freely available and commercial tools. Yet even the most basic operations like the multiplication of two matrices or computing a determinant can be excessively time-consuming and inaccurate if performed recklessly. This chapter elucidates the crucial aspects of matrix manipulation and standard algorithms for the classes of matrices most frequently encountered by a physicist, with particular attention to the analysis of errors and condition estimates. Singular value decomposition is illustrated in the framework of image compression. A separate section is devoted to random matrices occurring in the study of quantum chaos, information theory and finance, with emphasis on Gaussian orthogonal and unitary ensembles and their cyclic counterparts. The Examples and Problems include the calculation of energy states of particles in one-dimensional and two-dimensional potentials, percolations in random lattices, electric circuits of linear elements, Anderson localization, and spectra of symmetric random matrices.KeywordsCondition NumberSingular Value DecompositionRandom MatriceHermitian MatriceSparse MatriceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.