**Unformatted text preview: **= 1, 2, . . . , n, or, by using the identity 1 − cos θ = 2 sin2 (θ/2),
λij = 4 sin2 iπ
2(n + 1) + sin2 jπ
2(n + 1) , D
E i, j = 1, 2, . . . , n. (7.6.8) Since each λij is positive, L must be positive deﬁnite. As a corollary, L is
nonsingular, and hence Lu = g yields a unique solution for the steady-state
temperatures on the square plate (otherwise something would be amiss). T
H At ﬁrst glance it’s tempting to think that statements about positive deﬁnite
matrices translate to positive semideﬁnite matrices simply by replacing the word
“positive” by “nonnegative,” but this is not always true. When A has zero
eigenvalues (i.e., when A is singular) there is no LU factorization, and, unlike the
positive deﬁnite case, having nonnegative leading principal minors doesn’t insure
that A is positive semideﬁnite—e.g., consider A = 0 −0 . The positive
0
1
deﬁnite properties that have semideﬁnite analogues are listed below. IG
R Y
P Positive Semideﬁnite Matrices
For real-symmetric matrices such that rank (An×n ) = r, the following
statements are equivalent, so any one of them can serve as the deﬁnition
of a positive semideﬁnite matrix.
• xT Ax ≥ 0 for all x ∈ n×1 (the most common deﬁnition). (7.6.9) O
C •
• All eigenvalues of A are nonnegative.
A = BT B for some B with rank (B) = r. • All principal minors of A are nonnegative. For hermitian matrices, replace ( )T by ( )∗ and (7.6.10)
(7.6.11)
(7.6.12)
by C . Proof of (7.6.9) =⇒ (7.6.10). The hypothesis insures xT Ax ≥ 0 for eigenvectors
2
2
of A. If (λ, x) is an eigenpair, then λ = xT Ax/xT x = Bx 2 / x 2 ≥ 0.
Proof of (7.6.10) =⇒ (7.6.11). Similar to the positive deﬁnite case, if each λi ≥ 0,
write A = PD1/2 D1/2 PT = BT B, where B = D1/2 PT has rank r. Copyright c 2000 SIAM Buy online from SIAM
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7.6 Positive Deﬁnite Matrices
http://www.amazon.com/exec/obidos/ASIN/0898714540 567 Proof of (7.6.11) =⇒ (7.6.12). If Pk is a principal submatrix of A, then
Pk = QT AQ = QT BT BQ = FT F| =⇒ Pk = FT F It is ill...

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