508 guarantees that every a c mm is unitarily similar

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Unformatted text preview: nting algebraic multiplicities. In 1852 J. J. Sylvester (p. 80) discovered that the inertia of A is invariant under congruence transformations. Sylvester’s Law of Inertia Let A ∼ B denote the fact that real-symmetric matrices A and B = are congruent (i.e., CT AC = B for some nonsingular C ). Sylvester’s law of inertia states that: A∼B = Copyright c 2000 SIAM if and only if A and B have the same inertia. 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 Proof. 77 569 Observe that if An×n is real and symmetric with inertia (p, j, s), then ⎛ A∼⎝ = ⎞ Ip×p ⎠ = E, −Ij ×j (7.6.15) 0s×s It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] because if {λ1 , . . . , λp , −λp+1 , . . . , −λp+j , 0, . . . , 0} are the eigenvalues of A (counting multiplicities) with each λi > 0, there is an orthogonal matrix P such that PT AP = diag (λ1 , . . . , λp , −λp+1 , . . . , −λp+j , 0, . . . , 0) , so C = PD, −1/2 −1/2 where D = diag λ1 , . . . , λp+j , 1, . . . , 1 , is nonsingular and CT AC = E. Let B be a real-symmetric matrix with inertia (q, k, t) so that ⎛ B∼⎝ = D E ⎞ Iq×q ⎠ = F. −Ik×k T H 0t×t If B ∼ A, then F ∼ E (congruence is transitive), so rank (F) = rank (E), and = = hence s = t. To show that p = q, assume to the contrary that p > q, and write F = KT EK for some nonsingular K = Xn×q | Yn×n−q . If M = R (Y) ⊆ n and N = span {e1 , . . . , ep } ⊆ n , then using the formula (4.4.19) for the dimension of a sum (p. 205) yields IG R dim(M ∩ N ) = dim M + dim N − dim(M + N ) = (n − q ) + p − dim(M + N ) > 0. Y P Consequently, there exists a nonzero vector x ∈ M ∩ N . For such a vector, x ∈ M =⇒ x = Yy = K and O C x∈N 0 y =⇒ xT Ex = 0T | yT F =⇒ x = (x1 , . . . , xp , 0, . . . , 0)T 0 y ≤ 0, =⇒ xT Ex > 0, which is impossible. Therefore, we can’t have p > q. A similar a...
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