Because of his work on the convergence of

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Unformatted text preview: eading principal minor is positive, then all principal minors must be positive because if Pk is any principal submatrix of A, then there is a permutation matrix Q such that Copyright c 2000 SIAM Buy online from SIAM Buy from 7.6 Positive Definite Matrices Pk is a leading principal submatrix in C = QT AQ = Pk σ (A) = σ (C) , we have, with some obvious shorthand notation, 559 , and, since It is illegal to print, duplicate, or distribute this material Please report violations to A ’s leading pm’s > 0 ⇒ A pd ⇒ C pd ⇒ det (Pk ) > 0 ⇒ all of A ’s pm’s > 0. Finally, observe that A is positive definite if and only if xT Ax > 0 for every nonzero x ∈ n×1 . If A is positive definite, then A = BT B for a 2 nonsingular B, so xT Ax = xT BT Bx = Bx 2 ≥ 0 with equality if and only if Bx = 0 or, equivalently, x = 0. Conversely, if xT Ax > 0 for all x = 0, then for every eigenpair (λ, x) we have λ = (xT Ax/xT x) > 0. Below is a formal summary of the results for positive definite matrices. D E T H Positive Definite Matrices For real-symmetric matrices A, the following statements are equivalent, and any one can serve as the definition of a positive definite matrix. n×1 • xT Ax > 0 for every nonzero x ∈ the definition). • All eigenvalues of A are positive. • A = BT B for some nonsingular B. While B is not unique, there is one and only one upper-triangular matrix R with positive diagonals such that A = RT R. This is the Cholesky factorization of A (Example 3.10.7, p. 154). • A has an LU (or LDU) factorization with all pivots being positive. The LDU factorization is of the form A = LDLT = RT R, where R = D1/2 LT is the Cholesky factor of A (also see p. 154). (most commonly used as IG R Y P O C • The leading principal minors of A are positive. • All principal minors of A are positive. For hermitian matrices, replace ( )T by ( )∗ and by C . Example 7.6.1 Vibrating Beads on a String. Consider n small beads, each having mass m, spaced at equal intervals of length L on a very tightly stretched string or wire under a tension T as depicted in Figure 7.6.1. Each bead is init...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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