Com 76 positive denite matrices

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Unformatted text preview: ositive (or perhaps just nonnegative)? To answer this, let’s deal with real-symmetric matrices—the hermitian case follows along the same lines. If A ∈ n×n is symmetric, then, as observed above, there is an orthogonal matrix P such that A = PDPT , where D = diag (λ1 , λ2 , . . . , λn ) is real. If λi ≥ 0 for each i, then D1/2 exists, so A = PDPT = PD1/2 D1/2 PT = BT B B = D1/2 PT , for D E and λi > 0 for each i if and only if B is nonsingular. Conversely, if A can be factored as A = BT B, then all eigenvalues of A are nonnegative because for any eigenpair (λ, x), T H 2 λ= Bx 2 xT Ax xT BT Bx = = 2 ≥ 0. Tx x xT x x2 IG R Moreover, if B is nonsingular, then N (B) = 0 =⇒ Bx = 0, so λ > 0. In other words, a real-symmetric matrix A has nonnegative eigenvalues if and only if A can be factored as A = BT B, and all eigenvalues are positive if and only if B is nonsingular . A symmetric matrix A whose eigenvalues are positive is called positive definite, and when the eigenvalues are just nonnegative, A is said to be positive semidefinite. The use of this terminology is consistent with that introduced in Example 3.10.7 (p. 154), where the term “positive definite” was used to designate symmetric matrices possessing an LU factorization with positive pivots. It was demonstrated in Example 3.10.7 that possessing positive pivots is equivalent to the existence of a Cholesky factorization A = RT R, where R is upper triangular with positive diagonal entries. By the result of the previous paragraph this means that all eigenvalues of a symmetric matrix A are positive if and only if A has an LU factorization with positive pivots . But the pivots are intimately related to the leading principal minor determinants. Recall from Exercise 6.1.16 (p. 474) that if Ak is the k th leading principal submatrix of An×n , then the k th pivot is given by Y P O C ukk = for k = 1, det (A1 ) = a11 det (Ak )/det (Ak−1 ) for k = 2, 3, . . . , n. Consequently, a symmetric matrix is positive definite if and only if each of its leading principal minors is positive . However, if each l...
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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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