Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL SPECIL FORMS OF GENERLIZED INVERSES OF ROW BLOCK MTRICES YONGGE TIN bstract. Given a row block matrix B this paper investigates the relations between the generalized inverse B andthe column block matrix B consisting of two generalizedinverses and B. The first step of the investigation is to establish a formula for the minimal rank of the difference B B the secondstep is to finda necessary andsufficient condition for B B to holdby letting the minimal rank be zero. Seven types of generalizedinverses of matrices are taken into account. Key words. Block matrix Generalizedinverse Minimal rank formula Moore-Penrose inverse Schur complement. MS subject classifications. 1503 1509. 1. Introduction. Let C m n denote the set of all m n matrices over the field of complex numbers. The symbols r()andr() stand for the conjugate transpose the rank the range (column space) and the kernel of a matrix C m n respectively; B denotes a row block matrix consisting of and B. For a matrix C m n the Moore-Penrose inverse of denoted by isthe unique matrix C n m satisfying the four Penrose equations (i) (ii) (iii) () (iv) (). For simplicity let E I m and F I n. Moreover a matrix is called an {i...j}-inverse of denoted by (i...j) ifitsatisfiesthei...jth equations. The collection of all {i...j}-inverses of is denoted by { (i...j) }. In particular {1}-inverse of is often denoted by. The seven frequently used generalized inverses of are (1) (12) (13) (14) (123) (124) and (134) which have been studied by many authors; see e.g. 1 2 5 11. The general expressions of the seven generalized inverses of can be written as (1.1) (1.2) (1.3) + F V + WE (12) ( + F V )( + WE ) (13) + F V Received by the editors 3 ugust 2005. ccepted for publication 30 October 2005. Handling Editor: Ravindra B. Bapat. School of Economics Shanghai University of Finance andeconomics Shanghai 200433 China (yongge@mail.shufe.edu.cn). 249
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL 250 Y. Tian (1.4) (1.5) (1.6) (1.7) (14) + WE (123) + F V (124) + W E (134) + F VE where the two matrices V and W are arbitrary; see 1. Obviously { (134) } { (13) } { } { (134) } { (14) } { } { (123) } { (12) } { } { (124) } { (12) } { } { (123) } { (13) } { } { (124) } { (14) } { }. One of the main studies in the theory of generalized inverses is to find generalized inverses of block matrices and their properties. For the simplest block matrix M B where C m n and B C m k there have been many results on its generalized inverses. For example the Moore-Penrose inverse of B can be represented as (1.8) B ( + BB ) B ( + BB ) which is derived from the equality M M (MM ). nother representation of B in Mihályffy 4 is B (I n + TT ) 1 ( BS ) T (I n + TT ) 1 ( BS )+S where S E B and T BF S. Moreover it can easily be verified by definition that { } ( BB ) (I m BB ) B B ( BB ) (I m BB { B ) } { B(B B) (I m ) (B B) (I m ) } { B } (E B ) B B (E B ) { B (124) } B(E B) (E B) { B (124) } hold true.
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL GeneralizedInverses of Block Matrices 251 It has been shown that the rank of matrix provides a powerful tool for characterizing various equalities for generalized inverses of matrices. For example Tian 9 established the following three equalities: ( ) (1.9) r B B B B r B +2r( B) r() r(b) ( ) r B (1.10) B r B BB ( ) r B (EB ) (1.11) (E B) r()+r(b) r B. Let the right-hand side of (1.9) be zero we see that B { B } if and only if r B r() +r(b) 2r( B). However the inequality r B r() + r(b) r( B) always holds. Hence r B r()+r(b) 2r( B)isequivalent to B 0. Let the right-hand side of (1.10) be zero we see that B B if and only if B 0. Let the right-hand side of (1.11) be zero we see that B (EB ) (E B) if and only if r B r() +r(b). Motivated by these results we shall establish a variety of new rank formulas for the difference B (i...j) (i...j) B (i...j) and then use the formulas to characterize the corresponding matrix equalities for generalized inverses. Some useful rank formulas for partitioned matrices due to Marsaglia and Styan 3 are given in the following lemma. Lemma 1.1. Let C m n B C m k C C l n and D C l k. Then (1.12) (1.13) (1.14) r B r()+r(i m )B r(b)+r(i m BB ) r r()+r C( I C n )r(c)+r ( I n C C ) B r r(b)+r(c)+r(i C 0 m BB )( I n C C ). If R(B) R() and R(C ) R( ) then B (1.15) r r()+r( D C C D B ). pplying (1.15) and ( ) (see Zlobec 12) to a general Schur complement D C B gives the following rank formula: r( D C B )r B (1.16) C r(). D
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL 252 Y. Tian This formula can be used to find the rank of any matrix expression involving the Moore-Penrose inverse. In fact the equalities in (1.9) (1.10) and (1.11) are derived from (1.16). Because generalized inverse of a matrix is not necessarily unique the rank of a matrix expression involving generalized inverses of matrices may vary with respect to the choice of the generalized inverses. Suppose f( 1... p )andg(b1...b q ) are two expressions consisting of generalized inverses of matrices. Then there are 1... p B1...B q such that f( 1... p )g(b1...b q ) if and only if min 1... p B 1...B q r f( 1... p ) g(b 1...B q )0. One of the simplest matrix expressions involving a generalized inverse is the Schur complement D C B. In Tian 10 a group of formulas for the extremal ranks of the Schur complement D C (i...j) B with respect to (i...j) are derived. The following lemma presents some special cases of these formulas. Lemma 1.2. Let C m n and G C n m. Then (1.17) (1.18) (1.19) (1.20) (1.21) min r( G )r( G ) min r( (12) G )max{ r( G ) r(g)+r() r(g) r(g) } (12) min r( (13) G )r( G ) (13) min r( (14) G )r( G ) (14) min r( (123) G )r( G )+r (123) G r G (1.22) min r( (124) G )r( G )+r G r G (124) (1.23) min r( (134) G) r( G )+r(g ) r( G) (134) r( G )r (1.24) r(). G Let the right-hand sides of (1.17) (1.24) be zero we can obtain necessary and sufficient conditions for G { } { (12) } { (13) } { (14) } { (123) } { (124) } { (134) } and G to hold. If and G in (1.17) (1.24) are taken as matrix products matrix sums and partitioned matrices then many valuable results can be derived from the corresponding rank formulas. Some other rank formulas used in the context are given below: (1.25) min r( BC )r B +r C r B C 0 where C m n B C m k and C C l n ; see Tian 8. In particular (1.26) min r( B )r B r() r( B B ).
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL GeneralizedInverses of Block Matrices 253 The formula for minimal rank of B 1 1 C 1 B 2 2 C 2 is min r( B 1 1 C 1 B 2 2 C 2 )r (1.27) C 1 + r B 1 B 2 +max{ r 1 r 2 } 1 2 C 2 where B1 r 1 r C 2 0 B2 r 2 r C 1 0 r r B1 B 2 r B 1 C C 2 0 0 1 0 C 2 0 B1 B 2 r B 2 C C 1 0 0 1 0 ; C 2 0 see Tian 6. The following is also given 6: B (1.28) min r( C B )r r r B +r(). C 0 C 2. Main results. We first establish a set of rank formulas for the difference B (i...j) B. Theorem 2.1. Let C m n B C m k and let M B G B. Then (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) min r( M G )r(m)+2r( B) r() r(b) M min r( M (12) G )r(m)+2r( B) r() r(b) M (12) min r( M (13) G )r(m)+2r( B) r() r(b) M (13) min r( M (14) G )r B BB M (14) min r( M (123) G )r(m)+2r( B) r() r(b) M (123) min r( M (124) G )r B BB M (124) min r( M (134) G )r B BB M (134) r( M G )r B BB. Hence (2.9) G {M } M G B 0.
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL 254 Y. Tian Proof. pplying (1.17) to the difference M G gives (2.10) where min r( M G )r( M MGM ) M It is shown in 9 that M MGM M ( + BB )M BB B. (2.11) r BB B r B +2r( B) r() r(b). Substituting (2.11) into (2.10) yields (2.1). pplying (1.18) to M (12) G gives min r( M (12) G )max{ r( M MGM ) r(g)+r(m) r(gm) r(mg) }. M (12) It is easy to verify that ( ) r(g) r B r(m) r(gm) r B B r(m) Hence r(mg)r( + BB )r BB r(m). min r( M (12) G )r( M MGM )min r( M G ). M (12) M pplying (1.19) to M (13) G gives where M MG M + BB B + B min r( M (13) G )r( M MG M ) M (13) B Hence (2.3) follows from (2.11). From (1.20) min r( M (14) G )r( GMM M ) M (14) where GMM M BB B. Hence r( GMM M )r BB B r BB B BB B BB B. r B BB establishing (2.4). Equalities (2.5) (2.7) can be shown by (1.21) (1.23) and details are omitted. Eq.(2.8) is (1.10).
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL GeneralizedInverses of Block Matrices 255 set of rank formulas for the difference B (i...j) B (i...j) are given below. Theorem 2.2. Let C m n and B C m k and let M B. Then ( ) min r M (2.12) B B 2r()+2r(B) 2r(M) ( ) min r M (12) (2.13) (12) B (12) B (12) 2r()+2r(B) 2r(M) ( ) min r M (13) (2.14) (13) B (13) B (13) r BB B ( ) min r M (14) (2.15) (14) B (14) B (14) 2r()+2r(B) 2r(M) ( ) min r M (123) (2.16) (123) B (123) B (123) r BB B ( ) min r M (124) (2.17) (124) B (124) B (124) 2r()+2r(B) 2r(M) ( ) min r M (134) (2.18) (134) B (134) B (134) r BB B. Hence (a) There are ( (12) (124) ) and B (B (12) B (124) ) such that ( ) M B M (12) B (12) M (124) B (124) (2.19) if and only if r(m) r()+r(b) i.e. R() R(B) {0}. (b) There are (13) ( (123) (134) ) and B (13) (B (123) B (134) ) such that ( ) M (13) M (123) M (134) B (13) if and only if B 0. Proof. From (1.1) M B B (123) M + F V 1 + W 1 E B + F B V 2 + W 2 E B M B F 0 V1 0 F B V 2 B (134) W1 E W 2 E B where V 1 W 1 V 2 and W 2 are arbitrary. From (1.26) ( ) min r M F 0 V1 W1 E (2.20) V 1V 2 B 0 F B V 2 W 2 E B { ( )} r 0 0 B M W1 E B B W 2 E B
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL 256 Y. Tian Let N (2.21) where ( r M ( r M B P 1 0 B B ) W 1 E B BW 2 E B 0 and P 2 0 W 1 E W 2 E B ). B B 0 B. Then applying (1.27) gives B min r( M N P 1 W 1 E P 2 W 2 E B ) W 1W 2 r M N P 1 P 2 +r M N +max{r 1 r 2 } M r 1 r N P 1 E B 0 M r 2 r N P 2 E 0 r r E E B M N P 1 P 2 r M N P 1 E E B 0 0 0 E B 0 M N P 1 P 2 r M N P 2 E E 0 0 0. E B 0 Substituting (1.8) into the eight partitioned matrices in (2.21) and simplifying by (1.12) (1.13) and (1.14) give us the following results: r M N P 1 P 2 r()+r(b) r M N m + r()+r(b) r(m) E E B M r N P 1 M m +2r() r(m) r N P 1 P 2 m + r() E B 0 E B 0 0 r M N P 1 E 0 M m + r() r N P 2 m +2r(B) r(m) E E B 0 0 M r N P 1 P 2 m + r(b) r M N P 2 E E 0 0 0 m + r(b). E 0 The details are quite tedious and therefore are omitted here. Substituting these eight results into (2.21) and then (2.21) into (2.20) yields (2.12). From (1.3) M (13) B (13) M + F V 1 B M F 0 V1 (2.22) + F B V 2 B 0 F B V 2 where V 1 and V 2 are arbitrary. pplying (1.26) to this expression gives ( ) ( ) min r M F 0 V1 N r 0 V 1V 2 0 F B V 2 0 B ( M N) B
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL GeneralizedInverses of Block Matrices 257 as required for (2.14). From (1.4) M (14) B (14) M + W 1 E (2.23) B + W 2 E B M In N 0 r( M N) r BB B (by (1.10)) W 1 E 0 Ik W 2 E B where W 1 and W 2 are arbitrary. pplying (1.26) to (2.23) and simplifying by (1.12) (1.13) and (1.14) yield (2.15). The details are omitted here. Observe that ( min r M (13) (13) B (13) B (13) ) ( min r M (123) (123) B (123) B (123) ) r( M N ) and ( ) min r M (13) (13) B (13) B (13) ( min r M (134) (134) B (134) B (134) ) r( M N ). pplying (1.10) and (2.14) to these two inequalities gives (2.16) and (2.18). From (1.6) M (124) M + W 1 E (2.24) B (124) M N B + B BW 2 E B 0 W 1 E 0 B W B 2 E B where W 1 and W 2 are arbitrary. pplying (1.27) to (2.24) and simplifying by (1.12) (1.13) and (1.14) yield (2.17). lso note that ( ) min r M B B min (12) B (12) ( r M ) (12) B (12) min (124) B (124) ( r M (124) pplying (2.12) and (2.17) to this gives (2.13). Some rank formulas for B (i...j) (i...j) are given below. B (i...j) B (124) Theorem 2.3. Let C m n and B C m k and let M B. Then ( ) min r M (2.25) M B B r()+r(b) r(m) ).
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL 258 Y. Tian ( ) min r M (13) (13) (2.26) M (13) (13) B (13) B (13) r(m)+2r( B) r() r(b) ( ) min M (14) (14) (2.27) r()+r(b) r(m) M (14) (14) B (14) r B (14) ( ) min r M (123) (123) (2.28) M (123) B (123) r(m)+2r( B) r() r(b). (123) B (123) Hence (a) The following statements are equivalent: (i) There are and B such that B {M }. (ii) There are (14) and B (14) (14) such that {M (14) }. B (14) (iii) R() R(B) {0}. (b) The following statements are equivalent: (i) There are (13) and B (13) (13) such that B (13) {M (13) }. (ii) There are (123) and B (123) (123) such that B (123) {M (123) }. (iii) M. B (iv) B 0. Proof. From (1.17) ( ) ( ) min r M (2.29) M B r M M B M. Expanding the difference on the right-hand side of (2.29) yields M M B M BB B BB B +BW 1 E B W 2 E B where W 1 and W 2 are two arbitrary matrices. pplying (1.27) to the right-hand side and simplifying by (1.12) (1.13) and (1.14) give us ( ) (2.30) min r M M B B M minr(bb B +BV 1 E B W 2 E B ) V 1V 2 r()+r(b) r(m). Combining (2.29) and (2.30) results in (2.25). Eqs.(2.26) (2.27) and (2.28) can be shown by (1.19) (1.20) (1.21) and (1.27). The details are omitted here.
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL To find ( min r M (12) (12) M (12) B (12) (12) B (12) GeneralizedInverses of Block Matrices 259 ) ( min M (124) (124) B (124) r M (124) ( min M (134) (134) B (134) r M (134) (134) B (134) ) (124) B (124) a formula for the minimal rank of + B 1 1 C 1 + B 2 2 C 2 + B 3 3 C 3 with respect to 1 2 and 3 is needed. However such a formula has not been established at present; see 7. Y In addition to B B may have generalized inverses like or B for some matrices and Y. Theorem 2.4. Let C m n B C m k C k m Y C n m and let M B. Then ( min r (2.31) M M ) M ( ) Y min r M M Y B M r B +r( B) r() r(b). Hence the following statement are equivalent: (a) There is such that {M }. Y (b) There is Y such that B {M }. (c) r B r()+r(b) r( B). Proof. pplying (1.25) and simplifying by elementary block matrix operations give us ( ) min r M M M min r( 0B B B B ) r B 0 B B B +r B 0 B r B B B B 0 r 0 0 0 0 B B B +r r B B 0 r B B r(b) r B ( I m )B r(b) r( B)+r(I m )B r(b) r B +r( B) r() r(b). )
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL 260 Y. Tian Equality ( ) min r M M B M r B +r( B) r() r(b) can be shown similarly. The equivalence of (a) (b) and (c) follows from (2.31). Theorem 2.5. Let C m n B C m k C k m Y C n m and let M B. Then ( ) ( ) min r Y (2.32) M M M min r M M YB B M 0. Hence (a) There are and such that {M }. Y (b) There are Y and B and such that B {M }. Proof. pplying (1.25) and simplifying by elementary block matrix operations give us ( ) min r (2.33) M M M min r( 0B B B B ) 0 B r B B B +r B 0 B r B B B B 0 r B B r(b) r B ( I m )B r(b) r( B)+r(I m )B r(b) r( B)+r B r() r(b). lso from (1.28) min r( B)r()+r(B) r B. Combining this with (2.33) leads to ( ) min r M M M 0. The second equality in (2.32) can be shown similarly. Minimal rank formulas for column block matrices can be written out by symmetry. They are left for the reader. Some other topics on minimal ranks of generalized inverses of row block matrices can also be considered for instance to find the minimal ranks of (i...j) B (i...j) (EB ) (i...j) 1 (E B) (i...j) and 1... p (i...j). (i...j) p
Electronic Journal of Linear lgebra ISSN 1081-3810 publication of the International Linear lgebra Society EL GeneralizedInverses of Block Matrices 261 and then to use the minimal ranks to characterize the corresponding matrix equalities. For 2 2 partitioned matrices it is of interest to establish minimal rank formulas for the differences (i...j) B C 0 (i...j) C (i...j) B (i...j) 0 (i...j) B (i...j) C (i...j) C D B (i...j) D (i...j). It is expected that the rank formulas obtained can be used to characterize the corresponding equalities for generalized inverses of 2 2 partitioned matrices. REFERENCES 1. Ben-Israel andt.n.e. Greville. Generalized Inverses: Theory and pplications. 2ndEdn. Springer-Verlag New York 2003. 2 S.L. Campbell andc.d. Meyer. Generalized Inverses of Linear Transformations. Corrected reprint of the 1979 original Dover Publications Inc. New York 1991. 3 G. Marsaglia andg.p.h. Styan. Equalities andinequalities for ranks of matrices. Linear and Multilinear lgebra 2:269 292 1974. 4 L. Mihályffy. n alternative representation of the generalizedinverse of partitionedmatrices. Linear lgebra ppl. 4:95 100 1971. 5 C.R. Rao ands.k. Mitra. Generalized Inverse of Matrices and Its pplications. Wiley New York 1971. 6 Y. Tian. Upper andlower bounds for ranks of matrix expressions using generalizedinverses. Linear lgebra ppl. 355:187 214 2002. 7 Y. Tian. The minimal rank completion of a 3 3 partial block matrix. Linear and Multilinear lgebra 50:125 131 2002. 8 Y. Tian. The maximal andminimal ranks of some expressions of generalizedinverses of matrices. Southeast sian Bull. Math. 25:745 755 2002. 9 Y. Tian. Rank equalities for block matrices andtheir Moore-Penrose inverses. Houston J. Math. 30:483 510 2004. 10 Y. Tian. More on maximal andminimal ranks of Schur complements with applications. ppl. Math. Comput. 152:675 692 2004. 11 H. Yanai. Some generalizedforms of least squares g-inverse minimum norm g-inverse and Moore-Penrose inverse matrices. Comput. Statist. Data nal. 10:251 260 1990. 12 S. Zlobec. n explicit form of the Moore-Penrose inverse of an arbitrary complex matrix. SIM Review 12:132 134 1970.