Representations and sign pattern of the group inverse for some block matrices

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1 Electronic Journal of Linear Algebra Volume 3 Volume 3 25 Article Representations and sign pattern of the group inverse for some block matrices Lizhu Sun Harbin Institute of Technology sunlizhu678876@26com Wenzhe Wang Harbin Engineering University @qqcom Changjiang Bu Harbin Engineering University buchangjiang@hrbeueducn Yimin Wei Fudan University ymwei@fudaneducn Baodong Zheng Harbin Institute of Technology zbd@hiteducn Follow this and additional works at: Recommended Citation Sun Lizhu; Wang Wenzhe; Bu Changjiang; Wei Yimin; and Zheng Baodong 25 "Representations and sign pattern of the group inverse for some block matrices" Electronic Journal of Linear Algebra Volume 3 pp DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository For more information please contact scholcom@uwyoedu

2 REPRESENTATIONS AND SIGN PATTERN OF THE GROUP INVERSE FOR SOME BLOCK MATRICES LIZHU SUN WENZHE WANG CHANGJIANG BU YIMIN WEI AND BAODONG ZHENG A B Abstract Let M be a complex square matrix where A is square When BCB Ω C B rankbc rankb and the group inverse of Ω AB Ω CB Ω exists the group inverse of M exists if and only if rankbc + A B Ω AB Ω π B Ω A rankb In this case a representation of M # in terms of the group inverse and Moore-Penrose inverse of its subblocks is given Let A be a real matrix The sign pattern of A is a + -matrix obtained from A by replacing each entry by its sign The qualitative class of A is the set of the matrices with the same sign pattern as A denoted by QA The matrix A is called S 2 GI if the group inverse of each matrix à QA exists and its sign pattern is independent of à By using the group inverse representation a necessary A Y and sufficient condition for a real block matrix 2 to be an S 2 GI-matrix is given Y 2 where A is square and 2 are invertible Y and Y 2 are sign orthogonal Key words Group inverse Moore-Penrose inverse Sign pattern S 2 GI-matrix AMS subject classifications 5A9 5B35 Introduction Let C m n and R m n be the sets of m n complex matrices and m n real matrices respectively For A C n n the group inverse of A is a matrix X C n n satisfying AXA A XAX X AX XA Received by the editors on August 3 23 Accepted for publication on September Handling Editor: Bryan L Shader School of Science Harbin Institute of Technology Harbin 5 PR China sunlizhu678876@26com zbd@hiteducn College of Science Harbin Engineering University Harbin 5 PR China @qqcom College of Science College of Automation Harbin Engineering University Harbin 5 PR China buchangjiang@hrbeueducn Supported by the National Natural Science Foundation of China No 379 and the Natural Science Foundation of the Heilongjiang Province No QC24C School of Mathematical Sciences Shanghai Key Laboratory of Contemporary Applied Mathematics Fudan University Shanghai 2433 PR China ymwei@fudaneducn Supported by the National Natural Science Foundation of China No

3 Representations and Sign Pattern of the Group Inverse for Some Block Matrices 745 It is well-known that the group inverse exists if and only if ranka ranka 2 ; in this case the group inverse is unique see [] As is customary we denote the group inverse of A by A # When A is nonsingular A # A For A C m n the matrix X C n m is called the Moore-Penrose inverse of A if AXA A XAX X AX AX and XA XA where A is the conjugate transpose of A Let A + denote the Moore-Penrose inverse of A It is well-known that A + exists and is unique see [9] Throughout this paper A Ω I AA + A Z I A + A and A π I AA # where I is the identity matrix There are many applications of the group inverse of matrices in algebraic connectivity and algebraic bipartiteness of graphs see [5 9] Markov chains see [9] and resistance distance see [6] In 979 Campbell and Meyer proposed the open problem of finding explicit formulas for the Drazin or group inverse of a 2 2 block A B matrix in terms of its subblocks where A and D are square see [9] C D At present the problem of finding explicit representations for the group inverse of A B have not been completely solved Recently the existence and the representations for the group inverse of block matrices were given under some conditions C see [ ] Let sgna be the sign of a real number a which is defined to be or + depending on a < a or a > The sign pattern of A R m n is a + - matrix obtained from A by replacing each entry by its sign denoted by sgna ie for matrix A a ij m n sgna sgna ij m n The qualitative class of the real matrix A is the set of the matrices with the same sign pattern as A denoted by QA see [23] For A R n n A is called an SNS-matrix if each à QA is nonsingular The matrix A is called an S 2 NS-matrix if A is an SNS-matrix and sgnã sgna for each à QA see [4] The matrix A Rn n is called an SGI-matrix if à # exists for each à QA If A is an SGI-matrix and sgnã# sgna # for each à QA then A is an S2 GI-matrix sometimes we say A has signed generalized inverse to indicate that A is an S 2 GI-matrix see [25] The sign pattern of matrix has important applications in the qualitative economics see [ ] The monograph of Brualdi and Shader introduces many resultson S 2 NS-matricessee[4] In 995Shadergaveadescriptionforthe structure of matrices with signed Moore-Penrose inverse see [22] In 2 Shao and Shan completely characterized the matrices with signed Moore-Penrose inverse see [23] In 24 Britz Olesky and Driessche researched the signed Moore-Penrose inverse for the matrices with an acyclic bipartite graph see [3] In 2 M Catral et al proved that a nonnegative matrix corresponding to a broom graph has a signed group inverse see [2] In 24 Bapat and Ghorbani gave some results on the zero

4 746 L Sun W Wang C Bu Y Wei and B Zheng pattern of the inverse of lower triangular matricessee [2] In [25] a real block matrix A B M was shown to be an SGI-matrix if sgnb sgnc and C has C signed Moore-Penrose inverse and M is an S 2 GI-matrix with an additional condition A B A In [7 26] some results on real block matrices with signed Drazin C inverse were given under the condition sgnb sgnc and other conditions A B Let M be a complex square matrix where A is square When C B BCB Ω Ω AB Ω rankbc rankb and the group inverse of CB Ω exists weobtainthegroupinverseofm existsifandonlyifrankbc +A B Ω AB Ω π B Ω A rankb In this case we give the representation of M # in terms of the group inverse and Moore-Penrose inverse of its subblocks By using this representation we give a A Y necessary and sufficient condition for a real block matrix 2 to be Y 2 an S 2 GI-matrix where A is square and 2 are invertible ỸỸ2 for each Ỹ i QY i i 2 2 Some lemmas Before our main results some lemmas on the group inverse of 2 2 block matrix and the matrix sign pattern are presented First we define the notion of sign orthogonality and introduce other notations A B Lemma 2 [5] Let M be a complex square matrix where A is C D nonsingular If the group inverse of S D CA B exists then i M # exists if and only if R A 2 +BS π C is nonsingular; ii If M # exists X Y then M # where Z W X AR A+BS # CR A Y AR A+BS # CR BS π AR BS # Z S π CR A+BS # CR A S # CR A W S π CR A+BS # CR BS π S # CR BS π S π CR BS # +S #

5 Representations and Sign Pattern of the Group Inverse for Some Block Matrices 747 A Lemma 22 [] Let M B exists if and only if A # exists and ranka rank M # A # B A # 2 C n n and let A C r r Then M # A If M # exists then B A B Lemma 23 [24] Let M C n n If A BD C is invertible then C D M M/D M/D BD D CM/D D +D CM/D BD where M/D D CA B The term rank of a matrix denoted by ρa is the maximal cardinality of the sets of nonzero entries of A no two of which lie in the same row or same column It is easy to see that ρa n when A is an invertible matrix of order n Let A be an m n matrix and let [m] {m} [n] {n} If S and T are the subsets of [m] and [n] respectively then A[S T] denotes the submatrix of A whose rows index set is S and columns index set is T If S [m] or T [n] we abbreviate A[S T] by A[: T] or A[S :] Let A ij N r A and N c A denote the ij entry of matrix A the number of rows and the number of columns of the matrix A respectively Definition 24 If sgnã B for all the matrices à QA B QB N c A N r B then the matrices A and B are called sign orthogonal Lemma 25 [23] Let A be a real matrix of order n such that ρa n and A is not an SNS matrix Then there exist invertible matrices A and A 2 in QA and integers p q with pq n such that A qpa 2 qp < In[23 Theorem42]ShaoandShangavearesultonsgnÃ+ B C+ forallmatrices à QA B QB and C QC By using the similar methods as [23 Theorem 42] we establish a result on sgnabc for all matrices à QA B QB and C QC Lemma 26 Let ABC be real matrices with N c A N r B and N c B N r C If sgna BC sgnabc for all matrices B QB then sgnã B C sgnabc for all matrices à QA B QB and C QC Proof Let D ABC Then 2 D ij N cb k 2 N rb k A ik B kk 2 C k2j

6 748 L Sun W Wang C Bu Y Wei and B Zheng Ifthereexistmatricesà QA B QBand C QCsuchthatsgnà B C sgnabc then there exist integers i j p p 2 q q 2 and p q p 2 q 2 such that sgna ip B pq C qj + For k 2 let sgna ip 2 B p2q 2 C q2j B k pq { ε B pq p p k q q k B pq otherwise where ε > p N r B and q N c B Clearly B B 2 QB and 22 sgnab C sgnab 2 C Let D AB C D 2 AB 2 C By 2 D ij D 2 ij N cb k 2 N cb k 2 N rb k N rb k When ε is sufficiently small A ik B kk 2 C k2j + ε A ip B pq C qj A ik B 2 kk 2 C k2j + ε A ip 2 B 2 p2q 2 C q2j sgnd ij sgna ip B pq C qj + sgnd 2 ij sgna ip 2 B 2 p2q 2 C q2j Thus sgnd ij sgnd 2 ij which contradicts 22 So sgnã B C sgnabc for all matrices à QA B QB and C QC 3 Main results In this section some results on the existence representations and sign pattern for the group inverse of anti-triangular block matrices are given A B Theorem 3 Let M C n+m n+m such that the group inverse C B Ω AB Ω of CB Ω exists Let Γ BC +A B Ω AB Ω π B Ω A where A C n n If BCB Ω and rankbc rankb then

7 Representations and Sign Pattern of the Group Inverse for Some Block Matrices 749 i M # exists if and only if rankγ rankb; X Y ii If M # exists then M # where Z W X JAGAH JAH GAH JAG+G+H +J Y Γ + B +JAGAΓ + B JAΓ + B GAΓ + B Z C CGAΓ + I +AGAH AH AG+CG 2 I AH W C CGAΓ + A GAΓ + B Γ + B CG 2 AΓ + B J B Ω AB Ω π B Ω AΓ + H Γ + A B Ω AB Ω π B Ω G B Ω AB Ω # Proof By the singular value decomposition see [] there exist unitary matrices U C n n and V C m m such that 3 UBV where is an r r invertible diagonal matrix and r rankb Let UAU A A 2 VCU C C 2 32 A 3 A 4 C 3 C 4 wherea C C r r ThenM Φ MΦ whereφ is a unitary matrix and 33 Hence if M # exists then M A A 2 C C 2 A 3 A 4 C 3 C 4 U V I I I I 34 M # Φ M #Φ Since BCB Ω rankbc rankb 3 and 32 imply that C 2 and C is invertible Partition 33 into the following form M A A 2 C A 3 A 4 C 3 C 4 : N N 2 N 3 N 4

8 75 L Sun W Wang C Bu Y Wei and B Zheng where N A C A2 N 2 It is easy to see that N A3 N 3 C 3 C A C A4 N 4 C 4 Calculations show that M/N N 4 N 3 N N 2 N 4 It follows from 3 and 32 that B Ω AB Ω CB Ω B Φ Φ Ω AB Ω Note that the group inverse of N 4 CB Ω exists so the group inverse of M/N exists Since N is invertible and the group inverse of M/N exists by Lemma 2 M # exists if and only if R N 2 +N 2 M/N π N 3 is invertible A # According to Lemma 22 it yields that M/N # 4 2 C 4 A # Calcu- 4 lations yield R N 2 +N 2 M/N π N 3 N 2 +N 2 I M/N M/N # N 3 A 2 +A 2 A 3 A 2 A 4 A # 4 A 3 + C A C A C Note that C is invertible so rankr rank rank A 2 +A 2A 3 A 2 A 4 A # 4 A 3 + C A C A C A 2 A 3 A 2 A 4 A # 4 A 3 + C A C Hence R invertible implies that A 2 A 3 A 2 A 4 A # 4 A 3 + C be invertible that is ranka 2 A 3 A 2 A 4 A # 4 A 3 + C r rankb By simple computations we have Γ BC +A B Ω AB Ω π B Ω A A 2 A 3 A 2 A 4 A # 4 A 3 + C Thus R is invertible if and only if rankγ rankb Applying Lemma 2 we get # X Ỹ M Z W

9 Representations and Sign Pattern of the Group Inverse for Some Block Matrices 75 where X N R KR N Ỹ N R KR N 2 M/N π N R N 2 M/N # Z M/N π N 3 R KR N M/N # N 3 R N W M/N π N 3 R KR N 2 M/N π M/N # N 3 R N 2 M/N π M/N π N 3 R N 2 M/N # + M/N # K N +N 2 M/N # N 3 By M # Φ M #Φ Φ X Ỹ Z W Φ From 3 we get BB + U I B + B V I U B π U I V B Ω V I U V So by the above BB + B π B + B B Ω and 32 it yields 36 N Φ U Φ V BB + ABB + B B + BC A C U V Similarly we have Φ N2 Φ Φ N 3 M/N BB Φ + AB Ω B Φ Ω ABB + B Z CBB + Φ B Ω AB Ω CB Ω

10 752 L Sun W Wang C Bu Y Wei and B Zheng Note that BCB Ω Since the group inverse of M/N exists Lemma 22 and 39 imply that Φ M/N # Φ Φ A # 4 2 C 4 A Φ # 4 3 U V A # 4 2 C 4 A # 4 U V G CG 2 where G B Ω AB Ω # Similarly B Φ M/N π Φ Ω AB Ω π B Ω 3 CB Ω G B Z It follows from and 3 that K Φ Φ N Φ Φ N2 +Φ M/N # BB + AGABB + +BB + ABB + B 32 B + BC Note that R A 2 +A 2 A 3 A 2 A 4 A # 4 A 3 + C A C A C Computations yield that the Schur complement of R is R/C A 2 A 3 A 2 A 4 A # 4 A 3 + C Since R/C is invertible by Lemma 23 we have R S S A C A S C + A S A C where S R/C Computation shows that S Φ 33 Φ Γ + N 3 Φ

11 34 Representations and Sign Pattern of the Group Inverse for Some Block Matrices 753 Φ S A C Γ + A Γ + AB + BC + Φ [ B + B C ]+ 35 Φ A S Φ B + A B + AΓ + Similarly we have Φ C + A S A C 36 Adding yields 37 R Φ B + AΓ + AB + BC + +B + B + BC + Φ Γ + Φ Γ + Γ + AB + BC + B + AΓ + B + AΓ + AB + BC + +B + B + BC + Substituting the equations and 37 into 35 gives that X Y M # Z W where X JAGAH JAH GAH JAG+G+H +J Y Γ + B +JAGAΓ + B JAΓ + B GAΓ + B Z C CGAΓ + I +AGAH AH AG+CG 2 I AH W C CGAΓ + A GAΓ + B Γ + B CG 2 AΓ + B J B Ω AB Ω π B Ω AΓ + H Γ + A B Ω AB Ω π B Ω

12 754 L Sun W Wang C Bu Y Wei and B Zheng A B For the matrix M C following result Corollary 32 Let M in Theorem 3 when C B we have the A B B where A C n n and B C n m If the group inverse of B Ω AB Ω exists and let Γ BB +A B Ω AB Ω π B Ω A Then i M # exists if and only if rankγ rankb; X Y i If M # exists then M # where Z W X JAGAH JAH GAH JAG+G+H +J Y Γ + B +JAGAΓ + B JAΓ + B GAΓ + B Z B Γ + +B Γ + AGAH B Γ + AH B Γ + AG W B Γ + AGAΓ + B B Γ + AΓ + B G B Ω AB Ω # J B Ω AB Ω π B Ω AΓ + H Γ + A B Ω AB Ω π B Ω Let be a nonsingular matrix If sgn Ỹ sgn Y and sgnỹ2 sgny 2 for all the matrices Q Ỹ QY and Ỹ2 QY 2 N c N r Y N c Y 2 N r then Y and Y 2 are called sign unique Theorem 33 Let N A Y 2 Y 2 be a real square matrix where A is square and 2 are invertible Y and Y 2 are sign orthogonal Then N is an S 2 GI-matrix if and only if the following hold: i Y and Y 2 2 are sign unique; I Y 2 2 ii U A 2 Y is S2 NS-matrix I Proof Let B 2 A B Y and C Then N C Y 2 Since and 2 are invertible Y and Y 2 are sign orthogonal we get that B Ω and B rankbc rankb By computing we get BCB Ω Ω AB Ω CB Ω and rankbc + AB Ω AB Ω π B Ω A rankbc rankb By Theo-

13 Representations and Sign Pattern of the Group Inverse for Some Block Matrices 755 rem 3 the group inverse of N exists and 38 where N # 2 2 Y X X 2 Y 2 2 X 3 X 4 X A 2 X 2 A 2 Y X 3 Y 2 2 A 2 X 4 Y 2 2 A 2 Y Next we show that the condition is necessary Since is invertible ρ N r Next we show that and 2 are SNS-matrices If is not an SNS-matrix By Lemma 25 there exist matrices Q such that qp qp < where the integers p q with pq N r Let A Y A Y N 2 N 2 2 Y 2 Y 2 Clearly N N 2 QN and N # N # 2 exist From 38 and qp qp < we have N # N ra+qpn # 2 N ra+qp < This is contrary to N being an S 2 GImatrix Thus we have is an SNS-matrix Similarly 2 is an SNS-matrix  Ŷ Therefore for each matrix N 2 QN we have Ŷ 2 where X X 3 N # 2 Ŷ Ŷ X X 2 X 3 X 4  2 X2 Ŷ2 2 Since N is an S 2 GI-matrix we have  2 X4 Ŷ2 2  2 Ŷ  2 Ŷ sgn sgn sgn 2 sgn 2 sgn 2 Ŷ sgn 2 Y sgnŷ2 2 sgny 2 2 sgn X sgnx sgn X 2 sgnx 2 sgn X 3 sgnx 3 sgn X 4 sgnx 4

14 756 L Sun W Wang C Bu Y Wei and B Zheng for all matrices Q 2 Q 2 Ŷ QY and Ŷ2 QY 2 Since sgn sgn sgn 2 sgn 2 both and 2 are S 2 NS-matrices Hence there exists permutation matrix P such that P 2 ii i 2 N r P 2 I A P T Let Q P and W Q NQ T Y P 2 It I Y 2 follows from Theorem 3 that 2 PT 2 PT P Y W # P P X P T P X 2 Y 2 2 PT P X 3 P T X 4 Note that N is S 2 GI-matrix Thus W is an S 2 GI-matrix So sgn 2 PT P Ỹ sgn 2 PT P Y for all the matrices Q 2 Q 2 and Ỹ QY Next we prove Y and Y 2 2 are sign unique If Y is not sign unique Let Z P T and letz 2 P 2 Then Z Y P Y is not sign unique ie there exist integers i i 2 and matrices ỸỸ QY such that Z Ỹ ii 2 > and Z Ỹ ii 2 < For i N r Z 2 ε > let Z 2 [i :] { Z2 [i :] i i εz 2 [i :] i i Clearly Z 2 QZ 2 It is easy to see that Note that Z 2 i i { Z 2 i i i i ε Z 2 i i i i i N r Z 2 Z 2 Z Ỹ ii 2 ε Z 2 i i Z Ỹ ii 2 + N cz 2 ii i Z 2 i iz Ỹ ii2 Z 2 Z Ỹ ii 2 ε Z 2 i i Z Ỹ ii 2 + When ε is sufficiently small we get sgn Z 2 Z N cz 2 Z ii i Ỹ ii 2 sgn ε Z 2 i i Z Ỹ ii 2 2 i iz Ỹ ii2

15 Representations and Sign Pattern of the Group Inverse for Some Block Matrices 757 sgn Z 2 Z Ỹ ii 2 sgn ε Z 2 i i Z Ỹ ii 2 Since sgnz Ỹ ii 2 sgnz Ỹ ii 2 we have sgn Z 2 Z Ỹ ii 2 sgn Z 2 Z Ỹ ii 2 This contradicts the assumption that W is an S 2 GI-matrix So Y is sign unique and sgn 2 H sgn 2 Y for all matrices 2 Q 2 H Q Y Similarly Y 2 2 is sign unique and sgn H 2 sgny 2 2 for all matrices H 2 QY 2 2 Q Next we prove that part ii of the theorem holds Let Then L L 2 2 L 3 2 Y L 4 Y 2 2 X L AL 2 X 2 L AL 3 X 3 L 4 AL 2 X 4 L 4 AL 3 Since sgn  2 sgnx for all matrices Q  QA and 2 Q 2 wehavesgnl ÃL 2 sgnl AL 2 foreachmatrixã QA Itfollowsfrom Lemma 26 that sgn L  L 2 sgnl AL 2 for all matrices L QL  QA and L 2 QL 2 Similarly sgn L  L 3 sgnl AL 3 sgn L 4  L 2 sgnl 4 AL 2 sgn L 4  L 3 sgnl 4 AL 3 for all matrices  QA L QL L2 QL 2 L 3 QL 3 L 4 QL 4 Let U I Y 2 2 A 2 Y I Since and 2 are SNS-matrices U is an SNS-matrix By calculation we have 39 U I L 4 L 4 AL 2 L 4 AL 3 L L AL 2 L AL 3 L 2 L 3 I

16 758 L Sun W Wang C Bu Y Wei and B Zheng Clearly sgnû sgnu for all matrices Û Î Ĥ 2  2 Ĥ Î QU where Î Î QI Q 2 Q 2  QA Ĥ Q Y Ĥ 2 QY 2 2 Hence U is an S2 NS-matrix So i and ii hold If i and ii hold then by 38 and 39 N is an S 2 GI matrix A I Y Theorem 34 Let N I be a real square matrix where A is Y 2 square Y and Y 2 are sign orthogonal Then N is an S 2 GI-matrix if and only if sgnỹ2ã sgny 2A sgnỹ2ãỹ sgny 2 AY and sgnãỹ sgnay for each à QAỸ QY and Ỹ2 QY 2 Proof From Theorem 33 we have N is an S 2 GI-matrix if and only if U I Y 2 I A I Y is an S2 NS-matrix I Clearly U is an SNS-matrix Calculations gives U I Y 2 Y 2 A Y 2 AY I A AY I Y I Since sgnỹ2ã sgny 2A sgnỹ2ãỹ sgny 2 AY and sgnãỹ sgnay for each à QAỸ QY and Ỹ2 QY 2 we have U is an S 2 NS-matrix Hence N is an S 2 GI-matrix Acknowledgment The authors would like to thank the anonymous referee for his or her very hard work and giving some valuable comments which improved this manuscript significantly REFERENCES [] A Ben-Israel and TNE Greville Generalized Inverses: Theory and Applications John Wiley & Sons New York 974 [2] RB Bapat and E Ghorbani Inverses of triangular matrices and bipartite graphs Linear Algebra Appl 447:

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