Eigenvalues and Eigenvectors

Abstract

The decomposition of a matrix A into a product of two or three matrices can (depending on the characteristics of those matrices) be a very useful first step in computing such things as the rank, the determinant, or an (ordinary or generalized) inverse (of A) as well as a solution to a linear system having A as its coefficient matrix. In particular, decompositions like the QR, LDU, U′DU, and Cholesky decompositions—refer to Sections 6.4 and 14.5—in which the component matrices are diagonal or triangular, or have orthonormal rows or columns, can be very useful for computational purposes. Moreover, establishing that a matrix has a decomposition of a certain type can be instructive about the nature of the matrix. In particular, if it can be shown that a matrix A is expressible as A = RR′ for some matrix R, then it can be concluded—refer to Corollary 14.2.14—that A is nonnegative definite (and symmetric).