RANK REDUCTION AND BORDERED INVERSION

Transcription

1 Mathematical Notes, Misolc, Vol. 2., No. 2., (21), pp RANK REDUCTION AND BORDERED INVERSION Aurél Galántai Institute of Mathematics, University of Misolc 3515 Misolc Egyetemváros, Hungary [Received October 8, 21] Abstract. We clarify the connection between the ran reduction algorithm and the bordered inversion method and their related sequences. Mathematical Subject Classification: 15A9, 15A21, 15A23, 65F3, 65F99. Keywords: Ran reduction, bordered inversion, inverse factorization, conjugation 1. Introduction The ran reduction algorithm and the bordered inversion method play important roles in numerical linear algebra. The ran reduction algorithm, while eeping the size of the matrix, decreases its ran to zero. The bordered inversion method, starting from a small size leading principal submatrix, increases the size and ran of the submatrix to obtain an inverse. The formulae of the two methods show certain formal similarities. In this paper we are looing for a deeper connection between the two methods. Using unexplored observations of Egerváry [4], we determine three inds of relationship between the two algorithms and associated sequences. These results are essentially the following. The ran reduction operation gives the reverse bordered inversion formula. Using the inverse LDU decomposition algorithm included in the bordered inversion method, we can produce the same full ran factorization as the ran reduction algorithm. Finally, the bordered inversion formula leads to new multiplicative canonical forms of the ran reduction algorithm. 2. Preliminary results We need the following results. For proofs and other details, see Ouellette [12]. Lemma 1 (Guttman-1) Let E F A = R G H E R. (2.1) If E is nonsingular, then ran (A) =ran (E)+ran H GE F. (2.2)

2 118 A. Galántai Corollary 2 If A and E are nonsingular, then the Schur complement S =(A/E) =H GE F is also nonsingular. Lemma 3 (Guttman-2) Let E F A = G H R m n H R j j. (2.3) If H is nonsingular, then ran (A) =ran (H)+ran E FH G. (2.4) Corollary 4 If A and H are nonsingular, then the Schur complement is also nonsingular. T =(A/H) =E FH G 3. The inverse of partitioned matrices Here we recall the most important results concerning the inverse of partitioned matrices. For proofs and related results see Ouellette [12]. Let E F A = R n n E R (3.1) G H and E be nonsingular. Theorem 5 (Banachiewicz-Frazer-Duncan-Collar) If the nonsingular matrix A R n n is partitioned in the form (3.1), where E has an inverse, then A can be expressed in the partitioned form A = E + E FS GE E FS S GE S. (3.2) Remar 6 Provided that E is invertible, A is invertible if and only if S is invertible. Remar 7 Formula (3.2) can be written in the following form A = E + E F I S GE, I. (3.3)

3 Ran reduction and bordered inversion The bordered inversion method Formula (3.2) is the basis for the bordered inversion or escalator method of Frazer, Duncan and Collar [6]. For =1,...,m let E +1 = E F G H E R n n, F,G T R n p, H R p p. Hence n +1 = n +p and E m+1 = E. IfE and S = H G E F are nonsingular, then E +1 is also nonsingular and by Theorem 5 we can calculate E +1. The bloc bordered inversion algorithm and 1. Determine E For =1,...,m set E +1 = E S = H G E F + E S F S G E G E E F S S. (4.1) end It is clear that Em+1 = E provided that S is nonsingular for =1,...,m.IfS is singular for an index, then the Guttman lemma implies that E +1 is singular. Theorem 8 (Guttman) Let E be nonsingular. The bloc bordered inversion method gives the inverse of E if and only if all E are nonsingular for =1,...,m. Guttman suggested several bloc variants of the bordered inversion method. For details, see Guttman [1] and Ouellette [12]. We also note that Gergely [9] observed the equivalence of the bordered inversion method with the Gauss-Jordan inversion. 5. The ran reduction algorithm The ran reduction procedure of Egerváry is based on the following result. Theorem 9 (Egerváry-Guttman-Wedderburn) Let H, S R m n and ran (H) ran (S) =. Furthermore let UR V T (U R m, R R, V R n )bea full ran factorization of S. Then ran (H S) =ran (H) ran (S), (5.1) if and only if U = HX, V T = Y T H, Y T HX = R (5.2) for some matrices X R n and Y R m.

4 12 A. Galántai Remar 1 We can write that ran (H S) =ran (H) ran (S) S = HX Y T HX Y T H. For the proof of the theorem see Egerváry [4], Guttman [11], Cline and Funderlic [3], Ouellette [12] and Galántai [7]. The ran reduction algorithm which is important in full ran factorizations and conjugation has the following form. Let A 1 = A, X i R n l i, Y i R m l i, l i 1 and Yi T A ix i be nonsingular for i =1, 2,...,. The ran reduction procedure A i+1 = A i A i X i Y T i A i X i Y T i A i (i =1, 2,...,), (5.3) where ran (A) P i=1 l i. The algorithm is said to be breadown free, if all Yi T A ix i are nonsingular for i =1, 2,...,.Assumethat X ran (A) = l i. In this case, A +1 =.Let i=1 X =[X 1,...,X ], Y =[Y 1,...,Y ] and X (i) =[X 1,...,X i ], Y (i) =[Y 1,...,Y i ]. Then the following theorem is true. Theorem 11 The ran reduction procedure can be carried out breadown free if and only if Y T AX = Yi T AX j i,j=1 is bloc strongly nonsingular1.inthiseventtheran reduction procedure has the canonical form A i+1 = A AX (i) ³ Y (i)t AX (i) Y (i)t A (i =1,...,). (5.4) For proof, see Galántai [7], [8]. For i = the relation A = AX Y T AX Y T A holds, where Z = X Y T AX Y T is a reflexive inverse of A. Itisalsoeasytosee that algorithm (5.3) leads to the full ran factorization A = QD P T, (5.5) 1 It means that the matrix has a bloc LU decomposition without pivoting.

5 Ran reduction and bordered inversion 121 where P = A T 1 Y 1,...,A T Y, Q = A1 X 1,...,A T X and D = diag Y1 T A 1 X 1,...,Y T A X. If a square matrix B has a bloc LDU decomposition with respect to a given partition, then this unique factorization will be denoted by B = L B D B U B.Thenwe can prove Theorem 12 Let Y T AX be bloc strongly nonsingular. The components of the full ran factorization (5.5) are P = A T YL T Y T AX, and A can be expressed in the form A = AXU Y T AX For proof, see Galántai [7], [8]. Q = AXUY T AX, D = D Y T AX, (5.6) D Y T AX L Y T AX Y T A. (5.7) 6. Ran-reduction and bordered inversion I We show a surprisingly simple connection between the ran reduction operation and the bordered inversion. Theorem 13 Assume that E F A = R n n E R (6.1) G H and Then provided that H is invertible 2. and J = I n A AJ J T AJ J T A = R n (n ). (A/H) Proof. By simple calculation we have AJ J T AJ J T F A = H H FH [G, H] = G F G H A J J T AJ J T A =, E FH G which was to be proved. 2 This condition is in agreement with the strong bloc nonsingularity of Y T AX. In fact, X 1 = Y 1 = J and Y1 T AX 1 = H.,

6 122 A. Galántai Remar 14 If J is replaced by J b I = provided that E is nonsingular. A A b J ³ bj T A b J bj T A =,thenwehave (A/E), (6.2) Remar 15 If A and H are replaced by A and S, respectively then we have A /S = E. This means that A A J J T A J J T A = E. (6.3) We just obtained that one ran reduction step on matrix A results in the inverse of the leading principal submatrix E. Relation (6.3) gives a direct connection between the ran reduction and the bordered inversion. It was first discovered by Egerváry [4] in the scalar case (H R), who developed it for solving difference equations with modified boundary conditions [5]. The same idea appears in Brezinsi et al. [2], who give credit to Duncan and call it reverse bordered inversion. Their variant of identity (6.3) is obtained in the following way. Let " # be b A F = bg H b. Then be = E + E FS GE, F b = E FS, bg = S GE, H b = S. As F b ³ bh = E F and F b ³ bh bg = E FS GE we have E = be E FS GE = be bf ³ H b G, b (6.4) which is exactly the reverse bordered inversion formula of Brezinsi et al. [2]. 7. The bordered inversion method and triangular factorization We associate sequences with the bordered inversion algorithm which produce the inverse LDU factorizations of the leading principal submatrices E. It is based on the following observation. Let E = LDU be the unique bloc LDU decomposition of E. If E is not partitioned, then we just tae L = U = I and D = E. IfA has the form (3.1), then the inverse formula (3.3) can be written as A = U E F I = U A D A L A, D S L GE I

7 Ran reduction and bordered inversion 123 which is the inverse bloc LDU decomposition of A. Thus we can define the following sequences related to the bordered inversion algorithm of Section 4. Let E = L D U denote the unique LDU decomposition of E for =1,...,m. Inverse LDU decomposition algorithm For =1,...,m set L +1 = L G E U I, D +1 = D S E F U +1 = I end It is clear that E +1 = U +1 D +1 L +1 and. E = U m+1 D m+1 L m+1. (7.1) The above algorithm, which requires no extra operations, is based on a remar by Egerváry [4], according to which the bordered inversion (or escalator) method furnishes the triangular factorisation of the inverse. The associated procedure can be carried out if and only if the bordered inversion can be so. Hence the necessary and sufficient condition for performing the algorithm without breadown is the bloc strong nonsingularity of E., 8. Ran-reduction and bordered inversion II Here we determine a new relationship between ran reduction and bordered inversion. More precisely, we show that certain sequences related to the ran reduction and the bordered inversion algorithms, respectively, can be tied together. We recall the facts that 1. The associated sequences of the bordered inversion method produce U and L E. 2. The ran reduction method produces the full ran factorization A = QD P T = AXU Y T AX D Y T AX L Y T AX Y T A. E, D E If X = B T V, then the pair (P, V ) is bloc B-conjugate (see Galántai [7], [8]). This conjugation property plays an important role in the ABS methods (see Abaffy- Spedicato [1]). Having the two observations above, we can easily derive the following Conjugation algorithm 1. Apply the inverse LDU decomposition algorithm to the matrix Y T AX and get L Y T AX.

8 124 A. Galántai 2. Set P = A T YL T Y T AX. end Thus we have a new connection between ran reduction and bordered inversion. It is also clear that, similarly, we can have Q and D. Finally we mention that Stewart [13] gave a conjugation algorithm which is based on LU factorization. 9. Ran-reduction and the reverse bordered inversion We deal with the last connection found, which is, in fact, an application of identity (6.3) to obtain new canonical forms for A i+1. We start with the formula (5.4), which tellsusthat A i+1 = A AX (i) ³ Y (i)t AX (i) Y (i)t A (i =1,...,). Let I (j) be the first j columns of the identity matrix I n.furthermoreletω i = P i j=1 l j. Then X (i) = XI (ωi) and Y (i) = YI (ωi) and we can write Y A i+1 = AX T AX ³ I (ω i ) Y (i)t AX (i) I (ω i )T Y T A. As ³ I (ωi) Y (i)t AX (i) I (ω i)t = Y (i)t AX (i) we have the following Theorem 16 A i+1 has the following canonical form µ Y A i+1 = AX T AX Y (i)t AX (i) under the conditions of Theorem 11. Y T A (9.1) We can observe that in the parentheses we have a variant of the identity (6.3) with Y T AX instead of A.Thus Y T AX Y (i)t AX (i) = Y T AX J hj T Y T AX J i J T Y T AX, where J = Hencebysimplecalculationswehave I n ωi. (9.2)

9 Ran reduction and bordered inversion 125 Theorem 17 A i+1 has the following canonical form A i+1 = Y T J J T X A Y T J J T X (9.3) under the conditions of Theorem 11. Assuming that Y T AX " be F b = bg H b # wecanobtainathirdformofa i+1. Theorem 18 A i+1 has the following canonical form under the conditions of Theorem 11. A i+1 = Y T b H X (9.4) These new multiplicative canonical forms indicate different properties of the ran reduction algorithm. The first canonical form indicates the decrease of A +1.The second form shows the projector properties of the ran reduction matrix A i+1. The last canonical form shows the structure of A +1 in terms of inverse quantities. These three new forms certainly give a better understanding of ran reduction and may lead to further new results. Acnowledgement: This wor was supported by the CNR International short-term mobility program, Year 21. REFERENCES [1] Abaffy J. and Spedicato, E.: ABS-projection Algorithms: Mathematical Techniques for Linear and Nonlinear Algebraic Equations, Ellis Horwood, Chichester, [2] Brezinsi, C., Morandi Cecchi, M. and Redivo-Zaglia, M.: The reverse bordering method, SIAM J. Matrix. Anal. Appl., 15, (1994}, [3] Cline, R.E. and Funderlic, R.E.: Theranofadifference of matrices and associated generalized inverses, Lin. Alg. Appl., 24, (1979), [4] Egerváry, E.: On ran-diminishing operations and their applications to the solution of linear equations, ZAMP, XI., (196), [5] Egerváry, J.: Über eine Methode zur numerischen Lösung der Poissonschen Differenzengleichung für beliebige Gebiete, Acta Mathematica Academiae Scientiarium Hungaricae, 11, (196), [6] Frazer, A., Duncan, J., Collar, R.: Elementary matrices, Cambridge University Press, [7] Galántai A.: Ran reduction and conjugation, Mathematical Notes, Misolc, 1(1), (2),

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