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We will use this important fact to prove the quotient formula. Case 3. Suppose M — I. Then LMR = LR is the product of a block lower triangular matrix and a block upper triangular matrix, and {LR)/a = {L/a){R/a) - L [a^] R [a^]. 13) A computation shows that for block lower triangular matrices Li and L2 {L,L2)/a = {L,/a){L2/a), 24 BASIC PROPERTIES OF THE SCHUR COMPLEMENT CHAP. 1 and for block upper triangular matrices jRi and R2 {R,R2)/a = {Ri/a){R2/a). 13), for any k and lower triangular matrix R {LL*)la = {L/a){Lya) = (L ^ j ) (L [a^])*.

Thus, W — UVU*, so W and V are similar and hence have the same sets of eigenvalues. We conclude that p{A) — p{B) and q{A) — q{B), and hence that In(A) = In(^). If A and B are *-congruent and singular, they have the same rank, so z{A) = z{B). Thus, if we set Ai = Ip{A) © {—Iq{A)) and Bi = Ip{B) ® {—Iq{B))-> the nonsingular matrices Ai and Bi are the same size and Ai 0 O^(^) and Bi 0 ^Z{A) are *-congruent: Ai 0 OZ{A) = G* (Bi 0 OZ{A)) G for some nonsingular G. Partition G — [Gij]^ -^^ conformally with Ai ^OZ(A)- 28 BASIC PROPERTIES OF THE SCHUR COMPLEMENT CHAP.

N — A:. 4 Let H he annxn positive semidefinite matrix and let H[a] be a k X k nonsingular principal suhmatrix of H, 1 < k < n. Then Xi{H) > Xi{H[a']) > Xi{H/a) > A,+^(iJ), i = 1, 2 , . . , n - A;. 13) Proof. Since H, H[a], and H[a^] are all positive semidefinite, we obtain Hla""] > iJ[a^] - H[a'',a]{H[a])-^H[a,a''] = H/a. 12). 5 Let H he an n x n positive semidefinite matrix and let a and a' he nonempty index sets such that a' d a d {1, 2 , . . , n } . If H\oi\ is nonsingular, then for every i = l , 2 , .