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Partitioned Matrices and the Schur Complement

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Partitioned Matrices and the Schur Complement P Partitioned Matrices and the Schur Complement P–1 Appendix P: PARTITIONED MATRICES AND THE SCHUR COMPLEMENT TABLE OF CONTENTS Page §P.1. Partitioned Matrix P–3 §P.2. Schur Complements P–3 §P.3. Block Diagonalization P–3 §P.4. Determinant Formulas P–3 §P.5. Partitioned Inversion P–4 §P.6. Solution of Partitioned Linear System P–4 §P.7. Rank Additivity P–5 §P.8. Inertia Additivity P–5 §P.9. The Quotient Property P–6 §P.10. Generalized Schur Complements P–6 §P. Notes and Bibliography...................... P–6 P–2 §P.4 DETERMINANT FORMULAS Partitioned matrices often appear in the exposition of Finite Element Methods. This Appendix collects some basic material on the subject. Emphasis is placed on results involving the Schur complement. This is the name given in the linear algebra literature to matrix objects obtained through the condensation (partial elimination) process discussed in Chapter 10. §P.1. Partitioned Matrix Suppose that the square matrix M dimensioned (n+m) × (n+m), is partitioned into four submatrix blocks as A B × × M = n n n m (P.1) (n+m)×(n+m) C D m×n m×m The dimensions of A, B, C and D are as shown in the block display. A and D are square matrices, but B and C are not square unless n = m. Entries are generally assumed to be complex. §P.2. Schur Complements If A is nonsingular, the Schur complement of M with respect to A is defined as M/A def= D − CA−1 B.(P.2) If D is nonsingular, the Schur complement of M with respect to D is defined as M/D def= A − BD−1 C.(P.3) Matrices (P.2) and (P.3) are also called the Schur complement of A in M and the Schur complement of D in M, respectively. These equivalent statements are sometimes preferable. If both diagonal blocks are singular, see §P.10. See Notes and Bibliography for other notations in common use. §P.3. Block Diagonalization The following block diagonalization forms, due to Aitken, clearly display the Schur complement role. If A is nonsingular, − −1 I0AB I A B = A0, −CA−1 I CD 0I 0M/A (P.4) −1 AB= I0A0 IAB . CD CA−1 I 0M/A 0I whereas if D is nonsingular − −1 / I BD AB I0= M D0, 0I CD −D−1 CI 0D (P.5) −1 / AB= IBD M D0 I0. CD 0I 0DD−1 CI All of these can be directly verified by matrix multiplication. P–3 Appendix P: PARTITIONED MATRICES AND THE SCHUR COMPLEMENT §P.4. Determinant Formulas Taking determinants on both sides of the second of (P.4) yields −1 I0 A0 IAB A0 det(M) = − = = det(A). det(M/A). (P.6) CA 1 I 0M/A 0I 0M/A since the determinant of unit triangular matrices is one. Similarly, from the second of (P.4) one gets det(M) = det(D). det(M/D). (P.7) It follows that the determinant is multiplicative in the Schur complement. Extensions to the case in which A or D are singular, and its historical connection to Schur’s paper [658] are discussed in [817]. §P.5. Partitioned Inversion Suppose that the upper diagonal block A in (P.1)is nonsingular. Then the inverse of M in partitioned form may be expressed in several ways that involve (P.2): −1 −1 −1 −1 AB I −A B A 0 I0 M = = − CD 0I 0 (M/A)−1 −CA 1 I − −1 ( / )−1 −1 = I A B M A A 0 0 (M/A)−1 −CA−1 I (P.8) −1 −1 A 0 A B − − = + (M/A) 1 [ −CA 1 I ] 00 I −1 + −1 ( / )−1 −1 − −1 ( / )−1 = A A B M A CA A B M A −(M/A)−1 CA−1 (M/A)−1 Suppose next that the lower diagonal block D in (P.1) is nonsingular. Then the form corresponding to the last one above reads − 1 −1 −1 −1 −1 AB (M/D) −(M/D) BA M = = − − − − (P.9) CD −D 1C (M/D)−1 D 1 + D 1 C (M/D)−1 BD 1 If both A and D are nonsingular, equating the top left hand block of (P.9) to that of the last form of (P.8) gives Duncan’s inversion formula (M/D)−1 = A−1 + A−1 B (M/A)−1 CA−1.(P.10) or explicitly in terms of the original blocks, (A − BD−1 C)−1 = A−1 + A−1 B (D − CA−1 B)−1 CA−1.(P.11) When n = m so that A, B, C and D are all square, and all of them are nonsingular, one obtains the elegant formula due to Aitken: −1 −1 − (M/D) (M/B) M 1 = (P.12) (M/C)−1 (M/A)−1 Replacing −D by the inverse of a matrix, say, T−1, leads to the Woodbury formula presented in Appendix D. If B and C are column and row vectors, respectively, vectors and D a scalar, one obtains the Sherman-Morrison inverse-update formula also presented in that Appendix. P–4 §P.8 INERTIA ADDITIVITY §P.6. Solution of Partitioned Linear System An important use of Schur complements is the partitioned solution of linear systems. In fact this is the most important application in FEM, in connection with superelement analysis. Suppose that we have the linear system Mz = r, in which the given coefficient matrix M is partitioned as in (P.1). The RHS r = [ pq]T is given, while z = [ xy]T is unknown. The system is equivalent to the matrix equation pair Ax+ By= p, (P.13) Cx+ Dy= q. If D is nonsingular, eliminating y and solving for x yields x = (A − BD−1 C)−1 (p − BD−1q) = (M/D)−1 (p − BD−1q). (P.14) Similarly if A is nonsingular, eliminating x and solving for y yields y = (D − CA−1 B)−1 (q − CA−1p) = (M/A)−1 (q − CA−1p). (P.15) Readers familiar with the process of static condensation in FEM will notice that (P.14) is precisely the process of elimination of internal degrees of freedom of a superelement, as discussed in Chapter 10. In this case M becomes the superelement stiffness matrix K, which is symmetric. In the → → → = T notation of that Chapter, the appropriate block mapping is A Kbb, B Kbi , C Kib Kbi , D → Kii, p → fb, q → fi , x → ub, and y → ui . The Schur complement M/D → K/Kii is the ˜ = − −1 condensed stiffness matrix Kbb Kbb Kbi Kii Kib, while the RHS of (P.14) is the condensed ˜ = − −1 force vector fb fb Kbi Kii fi . §P.7. Rank Additivity From (P.4) or (P.5) we immediately obtain the rank additivity formulas rank(M) = rank(A) + rank(M/A) = rank(D) + rank(M/D), (P.16) or, explicitly in term of the original blocks AB − − rank = rank(A) + rank(D − BA 1C) = rank(D) + rank(A − CD 1B), (P.17) CD assuming that the indicated inverses exist. Consequently the rank is additive in the Schur comple- ment, and so it the inertia of a Hermitian matrix, as considered next. P–5 Appendix P: PARTITIONED MATRICES AND THE SCHUR COMPLEMENT §P.8. Inertia Additivity Consider the (generally complex) partitioned Hermitian matrix A11 A12 A = ∗ .(P.18) A12 A22 ∗ Here A11 is assumed nonsingular, and A12 denotes the conjugate transpose of A12. The inertia of A is a list of three nonnegative integers In(A) ={n+, n−, n0},(P.19) in which n+ = n+(A), n− = n−(A), and n0 = n0(A) give the number of positive, negative, and zero eigenvalues of A, respectively. (Recall that all eigenvalues of a Hermitian matrix are real.) Since A is Hermitian, its rank is n+(A) + n−(A), and its nullity n0(A). (Note that n0 is the dimension of the kernel of A and also its corank.) The inertia additivity formula in terms of the Schur complement of A11 in A,is In(A) = In(A11) + In(A/A11). (P.20) It follows that if A11 is positive definite, and A/A11 is positive (nonnegative) definite, then A is positive (nonnegative) definite. Also A and A/A11 have the same nullity. §P.9. The Quotient Property Consider the partitioned matrix . . A11 A12 B1 . AB . M = = A21 A22 B2 ,(P.21) CD ... ... ... ... C1 C2 . D in which both A and A11 are nonsingular. Matrix M need not be square. The quotient property of nested Schur complements is M/A = (M/A11)/(A/A11). (P.22) On the RHS of (P.22) we see that the “denominator” matrix A11 “cancels out” as if the expressions were scalar fractions. If M is square and nonsingular, then taking determinants and aplying the determinant formula we obtain det(M/A) = det(M/A11)/ det(A/A11). (P.23) §P.10. Generalized Schur Complements If both diagonal blocks A and D of M in (P.1) are singular, the notion of Schur complement may be generalized by replacing the conventional inverse with the Moore-Penrose generalized inverse. Further developments of this fairly advanced topic may be found in Chapters 1 and 6 of [817]. P–6 §P. Notesand Bibliography Notes and Bibliography Most of the following historical remarks are extracted from the Historical Introduction chapter of [817]. This edited book gives a comprehensive and up-to-date coverage of the Schur’s complement and its use as a basic tool in linear algebra. The name “Schur complement” for the matrices defined in (P.2) and (P.3) was introduced by Haynsworth in 1968 [345,346].
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