A square matrix is called a P-matrix if all its leading principal minors are positive. Sublasses of P-matrices very important in applications are the nonsingular totally nonnegative matrices and the nonsingular M-matrices. Other classes of P-matrices used for eigenvalue localization are also presented. We also present some recent results and applications of the following two classes of matrices: sign-regular matrices (which contains the class of totally nonnegative matrices) and H-matrices (which contains the class of M-matrices). Let us recall that nonsingular H-matrices are, in fact, strictly diagonally dominant matrices up to a column scaling. For diagonally dominant matrices and some subclasses of nonsingular totally nonnegative matrices, accurate methods for computing their singular values, eigenvalues or inverses have been obtained, assuming that adequate natural parameters are provided. We present some recent extensions of these methods to other related classes of matrices.
Juan Manuel Peña
Dept. of Applied Mathematics Universidad de Zaragoza, Spain