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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
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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 firstname.lastname@example.org 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
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
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
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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