Because of his work on the convergence of

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.6 Positive Definite Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 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 meyer@ncsu.edu 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...
View Full Document

This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online