EPr over F(X} AA+ A+A. For AeF, a generalized inverse. ON POLYNOMIAL EPr MATRICES

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1 Intenat. J. Hath. & Math. S. VOL. 15 NO. 2 (1992) ON POLYNOMIAL EP MATRICES 261 AR. MEENAKSHI and N. ANANOAM Depatment f Mathematics, Annamalai Univeslty, Annamalainaga- 68 2, Tamll Nadu, INDIA. (Received Hay 8, 1989) ABSTRACT. This pape gives a chaacteizatin f EP-X-matices. Necessay and sufficient cnditins ae detemined f {i} the Me-Pense invese f an EP matix t be an EP-X-matix and {ii} Me-Pense invese f the pduct f EP-X-matlces t be an EP-X-matix. Futhe, a cnditin f the genealized invese f the pduct f X-matices t be a X-matix is detemined. KEY WORDS AND PHRASES: EP-X-matlces, genealized invese f a matix. AMS SUBJECT CLASSIFICATION CODES" 15A57, 15A INTRODUCTION Let be the set f all mxn matices whse elements ae plymmtals in X ve an abitay field F with an invlutay autmphism a" a f a e F. The elements f F ae called x-matlces. F A(X} (alj{ X l} e F, A (X} (jt X}}- Let F be the set f all mxn matices whse elements ae atinal functins f the fm f( X}/g( X} whee f( X}, 8(X # ae plynmlals in X. F simplicity, let us dente A(X) by A itself. The ank f A ef is defined t be the de f its lagest min that is nt equal t the ze plynmial {[2]p.259}. AeF is said t be an unlmdula X-matix {} invetible in F if the deteminant f A{X}, that is, det A{X} is a nnze cnstant. AF is said t be a egula X-matix if and nly if it is f ank n {[2]p.259), that is, if and tly if the. kenel f A cntains nly the espectively [4]. We have unlmdula X-matices egula ze element. AeF is said t be EP ve the field.f(x) if k (A} and $ R(A) R(A whee R(A}and k (A) dente the ange space f A and ank f A x-matices} E P- X-matices }. Thughut this pape, let AeF]. Let 1 be identity element f F. The A + Me-Pense invese f A, dented by is the unique slutin f the fllwing set f equatins- AXA=A (1.1); XAX=X (1.2)" (AX)=AX [1.31; (XA} =XA ( A exists and A+F if and fly if k (AA*) k (A A) k (A) [7]. When A + exists, A is EP ve F(X} AA+ A+A. F AeF, a genealized invese (} {I invese is defined as a slutin f the plynmial matix equatin (1.11 and a eflexive genealized invese ) {1,2} invese is defined as a slutin f the equatins (1.1} and (1.2} and they belng t F. The pupse f this pape is t give a chaacteizatin f an EP X-matix. Sme esults n EP X-maices having the same ange space ae btained, As an applicatin necessay and sufficient cnditins ae deived f (AB) + t be an EP-X-matix wheneve A and B ae EP-X-matices.

2 262 AR. MEENAKSHI AND N. ANANDAM 2. CHARACTERIZATION OF AN EP -h-matrix THEOREM I. AEFn] is EP ve the field F{k) if and nly if thee exist an nxn unlmdula k-matix P and a x egula k-matlx E such that PAP D " PROOF. By the Smith s cannical fm, A =1 O whee P and O ae unimdula-k-matices f de n and D is a x egula diagnal h-matix. Any {1} invese f A is given by A (1) Q_1 D-11 p-1 whee R 2, R 3, and R 4 ae abitay cnfmable matices ve F(k). A is EP ve the field F(k) ==> QP QP % 1.., R3 ----@ R(A)= R(A =- *(1) A AA A * (By Theem 1713]) D R 3,-I, * *{ O O R 2 R 4 Patitining cnfmably, let, QP -1 T 3 T 4 D LT3 T 4 DT 1 DT2 =- T 2 (since D is egula). Theefe QP,-1 ITI! T 3 T4, + DT2R2D 1 DO T1 p, T1 * Hence A P P P P whee E DT 1 is a x egula h-matix. Cnvesely, let PAP E Since E is egula, E is EP ve F(X). =@ R(E) R(E == R(A) R(A = A b R(PAP R(PA P whee E is a x egula h-matix. is EP ve F(X). Hence the theem If A] and is EP ve the field F{k} then we can find nxn egula and K need nt be unlmdula h-matlces. F example, cnside A Ik{=.A is

3 POLYNOMIAL EP MATRICES 263 * -I E P, being a egula X-matix. If A HA then H A A If A" AK then - : -1 * il -1/X ae nt X-matices. K A A Hee H,XI unimdula and K ll/k 1 The fllwing theem gives a necessay cnditin f H and K t be X-matices. THEOREM 2. If A is an nxn EP-X-matix and A has a X-matix {1} invese then thee exist nxn unimdula X-matices H and K such that A HA AK. PROOF. Let A be an nxn EP -X-matix. By Theem 1, thee exists an nxn E unimdula X-matix P such that PAP whee E is a x egula -I x-matix. Since A has a X-matix {1 invese, E is als a X-matix. E Nw A p-1 p-1 I -I [ E* * Theefe A P P p-i E*E -I 1 E - pp-i I p-l* HA whee H p-i P is an nxn unimdula X-matix. Similaly we can wite A AK whee E-1E K P, p-1 is an nxn unimdula X-matix. Theefe A HA AK. REMARK I. The cnvese f Theem 2 need nt b tue. F example,,ix, A 1 Hweve A has n X-matix 1} invese. 3. MOORE-PENROSE INVERSE OF AN EP -X-MATRIX The fllwing theem gives a set f necesssy and sufficient cnditins f the existence f the X-matix Me-Pense invese f a given X-matix. cnside Since A A, H K 12 A is an EP1- X-matix. THEOREM 3. F A F], the fllwing statements ae equivalent. i) A is EP, k(a) k(a2) and A A has a X-matix {1 invese. ti) Thee exists an unimdula X-matix U with A U U, whee D is a x unimdula X-matix and U U is a diagnal blck matix. iii) A GLG whee L and G G ae x unimdula X-matices and G is a X-matix. iv) A + is a X-matix and EP, v) Thee exists a symmetic idemptent X-matix E, (E 2 E E such that AE EA and R(A) R(E). PROOF. (i) (ii) Since A is an EP X-matix ve the field F(X and k(a) k(a2), A / exists, by Theem 2.3 f [5]. By Theem 4 in [6], A A has a X-matix {1} invese implies that thee exists an unimdula X-matix P with pp whee P1 is a symmetic x unlmdula X-matix such that

4 264 AR. MEENAKSHI AND N. ANANDAM PA whee W is a xn, X-matix f ank. Hence by Theem 2 in [6], l AA is a k-matix and PAA+P * Pl Since A is EP AA + A+A and [ p-1!w p-1 P 1 A AA+A A(AA+). Theefe A P,. L" J H cnsists f the fist clumns f P thus H is a nx, X-matix f ank. Nw A p-i D p-i U U whee U p-i and D WH is a O x egula X-matix. Since A hasa X-matix {I} invese and P is an * unimdula X-matix, PAA P has a X-matix {I} invese. Theefe by Theem I in [6], D -ID is an unimdula X-matix D which implies D is an unimdula X-matix. Hence A U U whee D, O] is a x unimdula X-matix and U U is a diagnal blck X-matix. Thus (ii) hlds. (i) =- (iii) Let us patitin U as U U1 U2 u 3 u4 u 3 u4 Lu2 u whee U 1 is a x X-matix. Then GLG U 3 ae X-matices. Since U U is a dtasnal blck X-matix, G G UIU1 unimdula X-matices. Thus (iit) hlds. + + U3U3 and L ae x (iii) =) (iv) Since A GLG L and G G ae unimdula X-matices. One can veify that A + -- G(G* G) -IL-I (G*G)-IG*. Nw AA + GLG G (G G) (G G) G G(G G) G A+A implies that A + is RP Since L and G* * -1. G ae untmdula, L -1 and (G G) ae X-matices, and G is a X-matix. Theefe A + is a X-matix. Thus (iv) hlds. (Iv) (v) Pf is analgus t that f (II) (lil) f Theem 2.3 = [5]. {v) = {i) Since E is a symmetic Idemptent X-matix with R(A) R(E) and AE EA, by Theem 2.3 in [5] we have A is EP and k(a) k(a2) ==> A + exists. Since E + E and R(A) R(E) == AA EE + E. Nw AE EA (AA+)A A. Let e. and a. dente the jth clumns f E and A espectively. Then AE A =-- Aej aj, since ej is a A-matix, the equatin Ax aj whee aj is a X-matix, has a X-matix slutin. Hence by Theem 1 in [6] it fllws that A ha a X-matix {I} invese. Futhe AA + E is als a X-matix. Hence by Theem 4 in [6] we see that A A has a X-matix {1} invese. Thus (i) hlds. Hence the theem.

5 POLYNOMIAL EP MATRICES 265 REMARK 2. The cnditin (i) in Theem 3 cannt be weakened which can be seen by the fllwing examples. EXAMPLE I. Cnsl,dec thel 2XmJatlx2Xzl A I/ X A is EP 1 and k(a} k(a2} 1. A A 2 21 has X-matix 1} invese (since, L2X the tnvaiant plynmial f A A is X- which is nt the identity f F). F this A, A + I 1 1 is nt a X-matix. Thus the theem falls 4X 1 1 EXAMPLE 2. Cnside the matix A ve GF(5). A is EP I. Since k(a) k{a2), A A" has a X-matix. {I} invese (since any cnfmable X-matix is a X-matix {I invese}. F this A, A des nt exist. Thus the theem fails. REMARKS 3. Fm Theem 3, it is clea that if E is a symmetic idemptent X-matix, and A is a X-matix such that R{E} R{A} then A is EP z=> AE EA A + is a X-matix and EP. We can shw that the set f all EP-X-matices with cmmn ange space as that f given symmetic tdemptent X-matix fms a gup, analgus t that f the Theem 2.1 in [5]. COROLLARY 1. Let E E* E2 e F. Then H(E}={A e F: A is EP ve F(X} and R(A} R(E}} is s maximal subgup f F cntaining E as identity. PROOF. This can be pved simila t that f Theem 2.1 f [5] by applying Theem APPLICATION In geneal, if A and B ae X-matices, having X-matix {1} inveses, it is nt the X-matix 1} invese f bth A and B. But AB Since +2 X 3 tnvaiant plynmial f AB is 1+2 X 2 1, AB has n X-matix {1 invese. the The fllwin8 theem leads t the existence f X-matix {1} tnvese f the pduct AB. THEOREM 4. Let A, B F. If A 2 A and B has X-matix {1} invese and R(A}.C. R(B} then AB has a X-matix 1 invese. PROOF. Suppse ABx b, whee b ls a X-matix, is a cnsistent system. Then b R(AB) R{A) c._ R{B) and theefe Bz b. Since ]9 has a X-matix 1 invese, by Theem I in [6] we get z is a X-matix. Since A is idemptent, s in paticula A is a{1 }invese f A and b R{A}, we have Ab=b. Nw ABZ Ab b. Thus ABx b has a X-matix slutin. Hence by Theem1 in [6], AB has a X-matix 1} invese. Hence the theem. The cnvese f Theem 4 need nt be tue which can be seen by the fllwing example EXAMPLE 4. Let A B AB Hee X X

6 266 AR. MEENAKSHI AND N. ANANDAM R (A) R(B). Hence the cnvese is nt tue Next we shall discuss the necessay and sufficient cnditin f the Me-Pense invese f the pduct f EP-k-matices t be an EP-k-matix. THEOREM 5. Let A and B be EP-k-matices. Then A A has a k-matix {1} invese, k(a) k(a2) and R(A) R(B) if and nly if AB is EP and (AB) + B+A + is a k-matix PROOF. Since A and B ae EP with R(A) R(B) and k(a) k(a2), by a Theem f Katz [1], AB is EP Since A is a EP-k-matix k(a) k(a2) and A A has a k-matix {1 invese, by Theem 3 A + is a k-matix and thee exists a symmetic idemptent k-matix E such that R(A) R(E). Hence AA + AA + E. Since A and B ae EP and R(A) R(B), we have AA + BB + E A+A B+B. Theefe BE EB and R(B) R(E). Again fm Theem 3, f the EP -k-matix B, we see that B + is a k-matix Since A and B ae EP with R(A) R(B) we can veify that (AB) + B+A + Since B + and A + ae k-matices, it fllws that (AB) + is a k-matix. a k-matix I} invese and k(a} k{a2), Since AB and B ae EP R(E) R(AB) R((AB) R(B R(B) and k(ab) k(b) implies Cnvesely, if (AB) + is a k-matix and AB is EP then (AB) + is an EP-k-matix. Theefe by Theem 3 thee exists a symmetic idemptent k-matix E such that R(AB) R(E) and (AB) (AB) + E (AB) + (AB). Since k(ab) k(a) and R(AB)iR(A), we get R(A) R(E). Since A is EP by Remak 3, it fllws that A + is a EP-k-matix. Nw by Theem 3, A A has R(B) R(E). Theefe R(A) R(B). Hence the theem. REMARK 4. The cnditin that bth A and B ae EP-k-matices, is essential in Theem 5, is illustated as fllws" Let A and B A and B ae nt EP 1. A A has a k-matix 1} in vese and R(A) R(B). But AB is nt EP (AB) I is nt a k-matix. Hence the claim 1+4 k 2k ACKNOWLEDGEMENT The auths wish t thank the efeee f suggestins which geatly impved the pfs f many theems. REFERENCES I. KATZ l.j. Theems n Pducts f EP Matices II Lin Alg. Appl. I (1975), LANCASTER, P. and TISMENETSKY, M. They f Matices, 2nd ed., Academic Pess, MARSAGLIA, G. and STYAN, G.P.H Equalities and inequalities f anks f matices, Lin. and Multi. A1R. 2 (1974), MARTIN PEARL, On Nmal and EP Matices, Mich. Math. J. 6 (1959), MEENAKSHI, AR. On EP Matices with Enties fm an Abitay Field, Lin. and Multi. A_lg. 9 (198), M EENAKSHI, AR. and ANANDAM, N. Plynmial Genealized Inveses f Plynmial Matices, (submitted). 7. PEARL, M.H. Genealized Inveses f Matices with Enties Taken fm an Abitay Field, Lin. AIR. Appl. 1 (1968),

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