Semigroup of generalized inverses of matrices

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1 Semigroup of generlized inverses of mtrices Hnif Zekroui nd Sid Guedjib Abstrct. The pper is divided into two principl prts. In the first one, we give the set of generlized inverses of mtrix A structure of semigroup nd study some lgebric properties like fctoriztion nd commuttivity. We lso define n equivlence reltion in order to estblish n isomorphism between the quotient semigroup nd the semigroup of projectors on RA. In the second prt, we study reltion between semigroups ssocited to equivlent mtrices nd estblish correspondence between the set of mtrices nd the set of ssocited semigroups. We lso study some lgebric properties in this set like intersection nd prtil order. M.S.C. 2000: 15A09, 15A24, 20M15. Key words: Generlized inverse; equivlent mtrices; projectors; prtil order. 1 Introduction In the following, K represents the rel or the complex field. Let A be n m n mtrix over K. The generlized inverse, or the {1} inverse of A is n n m mtrix X over K, which stisfies the mtrix eqution AXA = A. If in ddition X stisfies the eqution XAX = X, then X is sid to be reflexive generlized inverse or {1, 2} inverse of A. It is known tht there re infinitely sets of {1} inverses nd {1, 2} inverses of mtrix A. Mny studies on generlized inverses nd their pplictions hve been done see [2], [5], [1]. Some of them re lgebric; like the sum see [4] nd the reverse order lw of product see [7]. In some res of binry mtrices, the usge of the generlized inverse of mtrix might provide results s well, like the CP property for binry mtrices see [5]. In this pper we will study two principl prts. In the first one, we will give the set of {1} inverses of mtrix A structure of semigroup nd study some lgebric properties like fctoriztion nd commuttivity. To study the property of fctoriztion in these semigroups, we will use the Moore-Penrose inverse. We will lso define n equivlence reltion in order to estblish n isomorphism between the quotient semigroup nd the semigroup of projectors on RA. The second prt concerns the reltion between semigroups ssocited to equivlent mtrices nd the Applied Sciences, Vol.12, 2010, pp c Blkn Society of Geometers, Geometry Blkn Press 2010.

2 Semigroup of generlized inverses of mtrices 147 correspondence between the set M m n K nd the set of the ssocited semigroups. Some lgebric properties in this set will be treted; like the intersection of semigroups nd isomorphisms. Also, ordering of sets in the rel inner product spce, is recent subject. In [7], this subject ddressed s min result, nd proved using the Lorentz-Minkowski distnce for ordering certin sets. We shll lso follow this pttern by using generlized inverses for ordering sets of the introduced semigroups by minus prtil order. 2 Preliminry notes Definition 2.1. Let A be n m n mtrix over K. The Moore-Penrose inverse of A denoted by A + is the n m mtrix X over K which stisfies the four equtions 2.1 AXA = A, XAX = X, AX = AX, XA = XA. For every mtrix there exists only one Moore-Penrose inverse. If X is t lest {1} inverse of A, then AX nd XA re projectors on RA nd RX the rnge spces of A nd X respectively nd rnkax = rnka = rnkxa. For more properties nd remrks, see [2], [1]. Denote by A {1} nd by A {1,2} the sets of ll {1} inverses nd {1, 2} inverses of A respectively. We will denote by smll letters the sub-mtrices of mtrix X nd by I nd 0 the identicl nd zero mtrices or identicl nd zero sub-mtrices. Lemms without proofs re either esy to proof or well known in the precedent references. Lemm 2.1. Let A be n m n mtrix over K of rnk r. Then 1 There exist non singulr mtrices P nd Q such tht A = Q 1 2 A + = P 1 1 Q. 3 The elements of A {1} re of the form P 1 1 r f A {1,2} re of the form P 1 1 r f e f r e Q. e g P. Q nd the elements of The first ssertion is n elementry lemm in liner lgebr. The second one nd the third need only direct verifiction in 2.1. For this purpose the proof ws omitted.

3 148 Hnif Zekroui nd Sid Guedjib 3 Min results 3.1 Semigroup on A {1} Fctoriztion nd commuttivity The min importnt point in the theorem below is fctoriztion of A {1,2}. For this purpose we introduce two clsses of A {1} : { } P A+ A = X A {1} /X = P 1 1 Q P AA + = { X A {1} /X = P 1 1 x 0 } Q. The nottions P A + A nd P AA + re justified by the fct tht these two clsses re the sets of fixed points under left nd right multiplictions by orthogonl projectors A + A nd AA +. In fct, for ny X A {1}, A + AX = X X = P 1 1 Q nd XAA + = X X = P 1 1 Q. x 0 Theorem 3.1. Let be lw on A {1} defined s follows: for ny X, Y A {1} ; X Y = XAY. Then 1 A {1} is semigroup. 2 For ny X, Y A {1} we hve X Y A {1,2} 3 A {1,2} is n idel of A {1}. 4 P AA + P A + A = A {1,2} nd P AA + P A + A = {A + }. 5 For ny X nd Y in A {1} we hve X Y = Y X AX = AY nd XA = Y A. Proof. 1 A XAY A = AXA Y A = AY A = A. Then X Y = XAY A {1}. The ssocitivity of is obtined from the ssocitivity of product of mtrices. So A {1} is semigroup. 2 Let X nd Y be in A {1}. Since A XAY A = A nd XAY A XAY = X AY A XAY = X AXA Y = XAY, we obtin X Y = XAY A {1,2}. 3 As A {1,2} A {1}, then 3 is just simple consequence of 2. 4 First, remrk tht P AA + nd P A + A re subsemigroups of A {1,2}. Let A = Q 1 P. For ny Z = P 1 1 Q A y y {1,2} there exist Y = P 1 1 Q P y 0 AA + nd X = P 1 1 Q P A + A such tht Y X = P Q y 0 1 = P 1 1 Q = Z. Consequently, we hve P AA + P A + A = A {1,2}. However, by direct computtion, we y y fined tht P A + A P AA + = {A + 1 }. For Z = p Q y y 1 P AA + P A + A we hve x = 0 nd y = 0. Then Z = P 1 Q 1 = A +. Therefore P AA + P A + A = {A + }. This ssertion cn be replced by uniqueness of fctoriztion of elements of A {1,2}. 5 For ny X nd Y in A {1} we hve XAY = Y AX AXAY =

4 Semigroup of generlized inverses of mtrices 149 AY AX AY = AX nd XAY = Y AX XAY A = Y AXA XA = Y A. Conversely, AY = AX XAY = XAX nd Y A = XA Y AX = XAX. Hence we hve X Y = Y X AX = AY nd XA = Y A. Note tht we hve not relly commuttivity on A {1}, but we hve kind of commuttivity modulo projectors. It is esy to prove tht the set of projectors of vector spce on subspce is semigroup under ordinry composition. As mny studies on projectors nd their properties hve been done, we will exploit this to mke properties of A {1} more esy. This clim will be reched by defining n equivlence reltion in A {1}, thus we will hve n isomorphism between the quotient semigroup of A {1} nd the semigroup of projectors on RA. It will cuse no confusion if we use the sme letter to designte projector nd its ssocited mtrix. Theorem 3.2. Let A M m n K, Π be the semigroup of projectors of K m on R A. Then we hve 1 For ny P Π, there exists X A {1} such tht AX = P. 2 There is n equivlence reltion on A {1} such tht under the quotient lw of, is semigroup isomorphic to Π. A {1} Proof. 1 Let P Π nd A A {1}. As AA is projector on R A, we hve AA P = P nd P A = A. Thus if we tke X = A P, we hve AXA = AA P A = AA A = A nd AX = P which mens tht Π = { AX/X A {1}}. 2 Let be reltion in A {1} defined by X Y AX = AY. Then it is esy to check tht is n equivlence reltion in A {1}. Let χ be the cnonicl mp from A {1} on A{1} A{1}. Then for every χx, χy, the quotient lw of is defined by χxχ Y = χ X Y. It esy to show tht A{1} is semigroup nd χ is n homomorphism. It immeditely follows tht there exists mp ψ from A {1} to Π defined by ψχx = AX. Now we check tht ψ is n homomorphism. Since A XAY = AXA Y = AY, we obtin XAY Y. Then χ XAY = χy. Therefore ψ χxχy = ψ χ X Y = ψ χ XAY = ψ χy = AY = AXA Y = = AX AY = ψ χ X ψ χy. Now, since AX = AY, it follows tht χx = χy. We conclude tht for ny AX Π there is unique χx A{1} such tht ψχx = AX which mens tht ψ is bijection. 3.2 The set A {1} of semigroups Denote by M {1} m nk the set M {1} m nk = { A {1} /A M m n K }. In this subsection, we will estblish reltion between M m n K nd M m nk {1} nd deduce some properties of M m nk {1} like isomorphism of semigroups, intersection of them nd reltion order. For this min, the following Lemm will be useful. Lemm 3.1. [2], [1]. Let A nd B be two equivlent mtrices, such tht B = Q 1 AP. Then, for every Y in B {1} there exists unique X in A {1} such tht Y = P 1 XQ.

5 150 Hnif Zekroui nd Sid Guedjib Isomorphism between semigroups Theorem 3.3. Let A nd B be two equivlent mtrices. Then A {1}, nd B {1}, re isomorphic. Proof. By using the previous Lemm, we cn define mp ϕ from A {1} on B {1} s follows ϕx = P 1 XQ. Then ϕ 1 is the inverse mp from B {1} on A {1} given by ϕ 1 X = P XQ 1. In ddition, for every X nd Y in A {1}, we hve ϕx Y = ϕxay = P 1 XAY Q = P 1 XQQ 1 AP P 1 Y Q = = ϕxbϕy = ϕx ϕy. Also, we hve for every X nd Y in B {1}, ϕ 1 X Y = ϕ 1 XBY = ϕ 1 XAϕ 1 Y = ϕ 1 X ϕ 1 Y. Then the mp ϕ is n isomorphism. We remrk tht ϕ A + = P 1 A + Q = B + only if P nd Q re orthogonl. Lemm 3.2. [4] Let A nd B be two mtrices. Then the following sttements re equivlent: rnka + rnkb A = rnkb. b Every {1} inverse of B is {1} inverse of both A nd B A. c R A R B = {0} nd R A t R B t = {0}. Theorem 3.4. There is one-to-one correspondence between M m n K nd M {1} m nk mps 0 to M n m K nd preserves isomorphisms between semigroups. Proof. Let ψ be mp from M m n K onto M m nk {1} defined for every A M m n K by ψa = A {1}. Since 0X0 = 0 for ny X M n m K, we get 0 {1} = M n m K. Thus ψ0 = M n m K. According to Lemm 3.2, if A {1} = B {1}, we hve rnka + rnkb A = rnkb nd rnkb + rnka B = rnka. Thus we hve rnk A B = 0 = rnk B A. Therefore A = B. Now, let A nd B M m n K such tht B = Q 1 AP. According to Lemm 3.1 nd Theorem 3.3, we hve B {1} = { P 1 XQ/X A {1}} = ϕ A {1}. Hence we hve ψb = ϕψa Intersection of semigroups Theorem 3.5. For every mtrix A M m n K there exists mtrix A M m n K such tht A {1} A {1}. b For ny mtrices A, B M m n K there exists n isomorphism ϕ from A {1} on ϕ A {1} such tht ϕ A {1} B {1}. Proof. Let rnka = r min m, n. It is sufficient to prove tht for A M m n K with rnka = r, there exists mtrix A M m n K such tht rnk A + A = rnka + rnka = min m, n. Applying Lemm 3.2, we hve every {1} inverse of A + A is {1} inverse of both A nd A. Thus A {1} A {1}. Let P nd

6 Semigroup of generlized inverses of mtrices 151 Q be two nonsingulr mtrices such tht A = Q 1 I P. Then it is sufficient to tke A = Q 1 P with w M 0 w m r n r K nd rnkw = min m, n r. b Let rnka = r, rnkb = s. According to 1, there exist mtrices A nd B M m n K of rnks min m, n r nd min m, n s such tht rnk A + A = rnka + rnka = min m, n nd rnk B + B = rnkb + rnkb = min m, n. It follows tht A + A {1} A {1} A {1}, B + B {1} B {1} B {1}. Since A + A, B + B hve the sme rnk, they re equivlent. So, there exists n isomorphism ϕ from A {1} on ϕ A {1} such tht ϕ A + A {1} = B + B {1}. Hence we hve ϕ A + A {1} ϕ A {1} A {1} = ϕ A {1} ϕ A {1} nd ϕ A + A {1} B {1} B {1}. Finlly, we hve ϕ A {1} B {1}. The nturl question rises whether it is possible to conclude tht for ny mtrices A, B M m n K we hve A {1} B {1}. The nswer is negtive. In fct if we tke B = αa for sclr α which is different of 1 nd 0, then if X A {1} B {1}, we hve αa = αaxαa = α 2 A. Thus α {0, 1} which contrdicts our ssumption. Consequently we hve A {1} B {1} = Prtil order in M {1} m n K Definition 3.1. [1] A minus prtil order, denoted by, is defined s follows for A, B M m n K, then A B if rnkb = rnka + rnk B A. Theorem The inclusion is prtil order in M m nk {} induced by the minus order in the reverse order. b Let = min m, n. For ny mtrix A M m n K of rnk r there exists sequence of mtrices A = A r A r+1... A m0 in M m n K such tht rnk A r = rnk A = r, rnka r+i = r + i for i = 1,... r. Thus, there exists sequence A {1} A {1} r = A {1} nd A {1} is the lst term. Proof. For A, B M m n K, if A B, then rnkb = rnka+rnk B A. By Lemm 3.2 it follows tht B {1} A {1}. Hence we hve the prtil order in M m nk. {1} b Let {v 1, v 2,...v r } be bsis of RA nd {v r+1, v r+2,...v m } be such together with the bsis of RA form bsis for K m. Let {e 1, e 2,...e n } be bsis for K n such tht A r e j = Ae j = v j for j = 1,...r nd A r e j = 0 for j = r + 1,...m nd for i = 1,... r, A r+i e j = v j for j = 1,...r + i nd A r+i e j = 0 for j = r + i + 1,...m. In these bses the mtrices A = A r nd A r+i re of the form A = A r = Q 1 I P, A r+i = Q 1 Ir+i 0 P for i = 1,...m 0 r. Then we hve rnk A r = rnka = r, nd rnk A r+i = rnk A r + i = r + i = rnka r + rnk A r+i A r,

7 152 Hnif Zekroui nd Sid Guedjib for i = 1,... r. Thus A = A r A r+1... A m0. By, we hve A {1}... A {1} r = A {1}. A {1} is the lst term becuse A m0 is of mximl rnk. 4 Conclusions In the present pper we give the set of generlized inverses of mtrix structure of semigroup nd fctorize its elements to simple ones. Unfortuntely, this semigroup is not commuttive. To get nice structurl result, we define n equivlence reltion nd we obtin n isomorphism between the quotient semigroup nd the semigroup of projectors. Furthermore we estblish one-to-one correspondence between the set of mtrices nd the set of ssocited semigroups. This correspondence preserves isomorphisms between semigroups nd mps the zero mtrix which is the zero of the dditive group of mtrices into the entire set of mtrices which is the unity relted to intersection of sets. Finlly, we hve proved tht for ny mtrix, there exists sequence of semigroups ordered by inclusion. References [1] R.B. Bpt, Liner Algebr nd Liner Models, Springer [2] Ben-Isrel nd T.T. Greville, Generlized inverses, theory nd pplictions, Kreiger Eds [3] O. Demirel, E. Soyturk, On the ordered sets in n-dimensionl rel inner product spces, Applied Sciences , [4] J.A. Fill nd D.E. Fishkind, The Moore-Penrose Generlized Inverse for Sums of Mtrices, SIAM J. Mtrix Anl. Appl , [5] M.Z. Nshed, Generlized Inverses, Theory nd Applictions, Acdemic Press [6] L. Noui, A note on consecutive ones in binry mtrix, Applied Sciences , [7] Z. Xiong, B. Zheng, The reverse order lws for {1, 2, 3} nd {1, 2, 4} inverses of two-mtrix product, Appl. Mth. Lett. 21, , Authors ddress: Hnif Zekroui nd Sid Guedjib Deprtment of Mthemtics, Fculty of Sciences, University of Btn, 05000, Btn, Algeri. E-mil: hnifzekroui@yhoo.fr, sidguedjib@yhoo.fr