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10) SEC. 2 THE QUOTIENT FORMULA 23 Now let J denote a square matrix whose entries are all 1. 10) shows that the Schur complement of A in the product / Y 0 \ f A J ) \ C B \ ( I D )\{) V J is 5J, where s denotes the sum of all entries of M/A. Of course, sJ is independent of Y and V and has rank 1. 10) shows that {LMR)/A is similar to M/A. Thus the eigenvalues of {LMR)/A can be obtained by computing those of M/A, and they do not depend on the choices of F , V, and the nonsingular matrix W. 10) shows that if a matrix N can be written as a product of a lower triangular matrix, a diagonal matrix, and an upper triangular matrix, say, A^ = CKU, then N/a^{C/a){K/a){U/a) is a factorization of N/a of the same form.

Now we are ready to present the following interlacing theorem . 2 Let H be an n x n Hermitian matrix and let A be a k x k nonsingular principal submatrix of H. Then for i == 1, 2 , . . , n — /c^ Ai(F^) > XiliH/A)^ > Xi+kiH^). 15) Proof. Let In(i7) = (p, q, z) and In(^) = (pi, qi, 0). 6. T n,-u+ein-y ^®* - ( ^ + '^'' ^, ^ C + sIn-k \ = f ^J ~\B* ^ C, in which e is such a small positive number that both Hs and A^ are nonsingular. Note that h\{He) = (p 4- 2:, g, 0), ln{A£) = In(A), and also In(i^) = ln{K^) for any Hermitian matrix K.

Suppose that its leading principal submatrix An is k x k and positive definite, and that A22 is negative semidefinite. If the last n — k columns of A are linearly independent, then A is nonsingular and ln{A) = {k, n~ k, 0). Proof. Let S be nonsingular and such that S^AnS P S 0 0 / = Ik] let 30 BASIC PROPERTIES OF THE SCHUR COMPLEMENT CHAP. 4i2)* {S*Ai2) is positive definite, so ln{A) = {k, 0, 0) + (0, n - /c, 0) = {k, n- k, 0). I The next theorem gives information about the inertia of bordered Hermit ian matrices.