 Advances in Linear Algebra & Matrix Theory, 2011, 1, 1-7 doi:10.4236/alamt.2011.11001 Published Online December 2011 (http://www.SciRP.org/journal/alamt) On Real Matrices to Least-Squares g-Inverse and Minimum Norm g-Inverse of Quaternion Matrices* Huasheng Zhang School of Mathematics Science, Liaocheng University, Shandong, China Email: zhsh0510@yahoo.com.cn Received November 20, 2011; revised December 24, 2011; accepted December 30, 2011 ABSTRACT Through the real representations of quaternion matrices and matrix rank method, we give the expression of the real ma-trices in least-squares g-inverse and minimum norm g-inverse. From these formulas, we derive the extreme ranks of the real matrices. As applications, we establish necessary and sufficient conditions for some special least-squares g-inverse and minimum norm g-inverse. Keywords: Extreme Rank; g-Inverse; Least-Squares g-Inverse; Minimum Norm g-Inverse; Quaternion Matrix 1. Introduction Throughout this paper, stands for the real number field, stands for the set of all m matrices over the quaternion algebra mnn22201 230123=== ,,,.aa iajakijkijkaaaa ==1, I, AT, A* and †A stand for the identity matrix’ the transpose’ the conjugate transpose and the Moore-Pen- rose inverse of a quaternion matrix A. In , for a qua-ternion matrix A,  m= dim.diAA dimA† is called the rank of a quaternion matrix A and denoted by .rAThe well known Moore-Penrose inverse A of mnA is defined to be the unique matrix mnX satisfying the following four Penrose equations 1) =,AXAA 2) =,XAXX 3) =,AXAX=. 4) XAXA A matrix X is called a least-squares g-inverse of A if it satisfies both 1) and 3) in the Penrose equations, and de-noted by a matrix X is called a minimum norm g-inverse of A if it satisfies both 1) and 4) in the Penrose equations, and denoted by The general expres-sion of 1,3 ;A1,31,4.AA and 1,4A can be written as 1,3 †=A,AALV (1) 1,4†=AAAWR (2) where , the two matrices V and W is a arbitrary; see [, pp. 44-46]. †=,ALIAA†=ARIAAFor convenience of representation, we suppose 01 23=AAAiAjAk (3) and 1,3 01 231,40123=,=ABBiBjBkACCiCjCk   (4) where 0123,,, ,mnAAAA 0123,,, ,mnBBBB 0123,,, .nmCCCC  For an arbitrary qu aternion matrix 12 34=,MMMiMjMk we define a map () from to by mn44mn432134 1221 431234=.MMMMMM MMMMM MMMMMM (5) By (5), it is easy to verify that () satisfies the fo llow-ing properties: a) ==.MNMN b) =,MNMN =,MNMN =,kMkM k. c) 11==mnmnMTMTRMR   =,mnSMS1 *This research was supported by the Natural Science Foundation of China (11001115). where Copyright © 2011 SciRes. ALAMT H. S. ZHANG 2 000000000 000=,=,00 000 000000 000000 0=,=,.000000tttttttttttttttIIIITRIIIIIIStmnII d)  =4 .rM rMThe least-squares g-inverse and minimum norm g-inverse have a very wide range of applications in nu-merical analysis and mathematical statistics and have been examined by many authors(see, e.g., [3-6]). Haruo  developed some equivalent conditions on least-squares general inverse in 1990. Tian  presented the maximal and minimal ranks of the Schur complement to least-squares g-inverse and minimum norm g-inverse in 2004. Tian  establish necessary and sufficient conditions for a matrix to be the least-squares g-inverse and minimum norm g-inverse from rank formulas in 2005. Guo, Wei and Wang  derived structures of least squares g-inverses and minimum norm g-inverse of a bordered matrix in 2006. Quaternion matrices play an important role in me-chanics, computer science, quantum physics, signal and color image processing and so on. More and more inter-ests of quaternion matrices have been witnessed recently (e.g. [7-14] ) . Noticing that the properties of the real matrices in least-squares general inverse 1,3A and minimum norm g-inverse 1,4A (4) have not been considered so far in the literature. We in this paper use the real representa-tions of quaternion matrices and matrix rank method to investigate (4) over . In Section 2, we first give the expression of the real matrices i and in (4), then determine the maximal and minimal ranks of the real matrices i and i in (4). As applications, we establish necessary and sufficient condi-tions for a quaternion matrix has a pure real or pure imaginary BB=0Ci =0,1,2,3iCi2,3,1,1,3A and 1,4A. The necessary and suffi-cient conditions for all 1,3A and 1,4A are pure real or pure imaginary of a quaternion matrix are also presented. 2. Main Results We begin with the following lemmas which proof just like those over the complex field. Lemma 2.1 (see ) Let A, B and C be matrices over . Then ,,mnmkln a)   ,= =ABrABrArRBrBrRA,,L b)  ==ACArrArCLrCrALC c)  =.0BCABrrBrCrRAC Lemma 2.2 (see ) Let ,mnA,mpBqnC be given. Then a) The maximal rank of ABXC with respect to X is max= min,;XArABXCrAB rC (6) b) The minimal rank of ABXC with respect to X is min =.0XAABrA BXCrABrrCC    (7) Theorem 2.3 Suppose  is a least- , =1,2,3,4ijXij44squares g-inverse of ,A where ij A and ,nmX1,3A are defined as (3 ) and (4). Then in (4) can be written as 1,2,3 =0,iBi0 112233441 1221433421331244341143221=,41=,41=,41=.4BXXXXBXXXXB XXXXBXXXX23 (8) Written in an explicit form, in (4) are  =0,1,2,3iBi    ††01 122††33441212343411=4411 44 ,,,,AAAABPAQPAQPAQ PAQVVPL PL PL PLVV (9)     ††11 22 1††4334211243 3411=4411 44 ,,,,AAAABPAQPAQPAQ PAQVVPL PL PL PLVV(10) Copyright © 2011 SciRes. ALAMT H. S. ZHANG 3   ††21 324††31423113 2 44211=4411 44 ,,,,AA AABPAQPAQPAQ PAQVVPLPL PL PLVV  (11)    ††31 42 3††3241411432 2311=4411 441 ,,,,4AAAABPAQPAQPAQ PAQVVPL PL PL PLVV (12) where 1234=,0,0,0 ,=0,,0,0 ,=0,0,,0,=0,0,0,,nnnnPIP IPIP I 1234=,0,0,0 ,=0,,0,0 ,= 0,0,,0,=0,0,0,,TTmmTTmmQIQ IQIQ I and V1, V2, V3 and V4 are arbitrary real matrices with compatible sizes. Proof. Suppose 44 , =1,2,3,4ijXij, is a least- squares g-inverse of AX where i.e. ,nmij  4444 44=,=.ijij ijAXAAAX AX   Then applying property (c) of () above to them yields 111441144 44=,=;mnijmnmmnij mnijTATXTATTATTATXTATX   n 111441144 44=,=;mnijmnmnmnij mnijRARXRAR RARRARX RARX    111441144 44=,=.mnijmnmnmnij mnijSASXSASSASS ASXS ASX   Hence,  1441144 44=,=;nij mnij mnij mAT XTAAATX TATX T   1441144 44=,=;nij mnij mnij mAR XRAAARX RARX R   1441144 44=,=,nij mnij mnij mAS XSAAASX SASX S   which implies that and 1144 44,nij mnijmTXT RXR  144nij mSXS are also least-squares g-inverses of A. Thus, 1144 444414414 ijnijm nijmnij mXTXT RXRSX S  is also a least-squares g-inverse of ,A where 1144 4444144 44=ijnijmnijmijnij mXTXTRXRSX SX     and 11 11 22 33 4412 12 21 43 3413 13 3124 4214 41 14322321 2112 34 4322 11 22 33 4423 41 14322324 2442133131 13 31 244232 41 143223=,=,=,=,=,=,=,=,=,=,XXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX33 3344 11 2234 2112344341 41 14322342 2442133143 12 21 43 3444 3344 11 22=,=,=,=,=,=.XXXXXXXXX XXXXXXXXXXXXXXXXXXXXX Copyright © 2011 SciRes. ALAMT H. S. ZHANG 4 Let 11 22 33 4412 21 43 341331 244241 1432231=41 41 41 ,4XXXXXXXXXiXXXXjXXXXk Then, by (5), 144 441144 441=4 ,ijnijmnijm nijmXXTXTRXR SXS   is a least-squares g-inverse of A. Hence, by the property (a) of () we know that ˆX a least-squares g-inverse of A. The above discussion shows that the least-squares g-inverse of A and the least-squares g-inverse of A are equivalent. Observe that ,ijX i, j = 1,2,3,4 in (8) can be written as ˆ=.iji jXPXQ From (1), the least-squares g-inverse of A can be written as †=AXALV where and 1234=,,,VVVVV 41234,,,pqVVVV are arbitrary. Substituting them into (8) yields the four real matrices B0, B1, B2 and B3 in (9)-(12). According to Lemma 2 and Theorem 2, we can get the following extreme ranks formulas for the real matrices in the least-squares g-inverses. Theorem 2.4 Suppose that A and 1,3A are defined as (3) and (4). Then a) 00max=min4,;rBrAArA nm 00min =.rBrAA rA b) 11max= min4,;rBrAArA nm 11min =.rBrA A rA c) 22max= min4,;rBrAArA nm 22min =.rBrAA rA d) 33max=min4,;rBrAArAnm 33min =,rBrAArA where 01 210 0012322333 10123103 20230 12103=,=,=,=000000=,=00 0000AAAAA AAAAAAAAAAAAA3012,AAAAAAAAAAAAAA AAAAAA ,,, and  0312120301 2321303021023 01313 21021220 12 31310 320013=,= ,= ,==,==AAAAAAAAAA AAAAAAAAAAAAA AAAAA AAA AAAAAAA AAAAA AAAAAAA       210323032 1.AA AAAAAA A Proof. Applying (6) and (7) to B0 in (9), we get the following 00max= min,;min =,0mmPrBrPP rIPPrBrP PrPrI where     ††112 2††334123411=4411 ,44=,,,AAAAPPAQPAQPAQPAQPPLPLPLPL4. By Lemma 1, it is not difficult to find that Copyright © 2011 SciRes. ALAMT H. S. ZHANG 51234123401230 0000=4000 000 0000 0 00000000=400 0000=4,rP PPP PPPArrAAAAAPPPPAAAArrAAAArAArA n  where 0123,,,AAAA and A are defined as above. By the same manner, we can get extreme ranks of B1, B2 and B3. As one of important applications of the maximal and minimal ranks to real matrices, Theorem 2 can help to get the necessary and sufficient conditions for the exis-tence of some special least-squares g-inverses. We show them in the following. Corollary 2.5 Suppose 0123=.AAAiAjAk  Then a) A quaternion matrix A has a real least-squares g-inverse if and only if 123===rAArAArAArA . b) All least-squares g-inverses of quaternion matrix A are real matrices if and only if 123===4rAArAArAArA n, where 123,,AAA and A are def i ned as Theorem 2. Corollary 2.6 Suppose 01 23=.AAAiAjAk  Then a) A quaternion matrix A has a pure imaginary least- squares g-inverse if an d only if 0=.rA ArA b) All least-squares g-inverses of quaternion matrix A are pure imaginary matrices if and only if 0=4 ,rAArA n  where 0A and A are def i ned as Theorem 2. The following several theorems of minimum norm g-inverse can be shown by a similar approach, and their proofs are omi tte d he re. Theorem 2.7 Suppose is a 44 , =1,2,3,4ijYijminimum norm g-inverse of ,A where A and ,nmijY1,4A are defined as (3) and (4) Then in (4) can be written as  =0,1,2,3iCi0 112233443442231122143213312434114321=,41=,41=,41=.4CYYYYCYYYYCYYYYCYYYY    Written in an explicit form,  =0,1,iCi 2,3 in (4) are   241234,QQRQRQRQRQ††01 12††334123411=4411 441 ,,,4AAAACPAQPAPAQPAWWW W  1342134,QQRQRQRQRQ††11 22††43124311=4411 441 ,,,4AAAACPAQPAPAQ PAWWWW    423124,AAAAQQRQRQRQRQ††21 32††31413 2411=4411 441 ,,,4CPAQPAPAQ PAWW W W  ††31 4232414 3211=4411 441 ,,,4AAAACPAQPAPAQ PRQRQWWWW RQRQ3††14123,QAQ Copyright © 2011 SciRes. ALAMT H. S. ZHANG 6 where 12341234=,0,0,0 ,=0,,0,0 ,= 0,0,,0,= 0,0,0,,=,0,0,0 ,=0,,0,0 ,= 0,0,,0,= 0,0,0,,nnnnTTmmTTmmPIP IPIP IQIQ IQIQ I and W1, W2, W3 and W4 are arbitrary real matrices with compatible sizes. According to Lemma 2 and Theorem 3, we can get the following extreme ranks formulas for the real matrices in minimum norm g-inverse. Theorem 2.8 Suppose that A and 1,4A are defined as (3) and (4). Then a) 00max= min4,4;ArCrrAm nA 00min =.ArC rrAA b) 11max= min4,4;ArCrrAmnA 11min =.ArC rrAA c) 22max= min4,4;ArCrrAmnA 22min =.ArC rrAA d) 33max=min4,4;ArCrrAm nA 33min =,ArC rrAA where 0012 31133 22230133210==,=,=,012323 01010 32012332 101103 2012332102230 10123321 0223 011032000000=,000000=,=,=,=AAAAAAAA AAAA AAAAA AAAA AAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAA A A. As one of important applications of the maximal and minimal ranks to real matrices, Theorem 4 can help to get the necessary and sufficient conditions for the exis-tence of some special minimum norm g-inverse. We show them in the following. Corollary 2.9 Suppose 01 23=.AAAiAjAk  Then a) A quaternion matrix A has a real minimum norm g-inverse if and only if 123===AAArrrrAAA        .A b) All minimum norm g-inverse of quaternion matrix A are real matrices if and only if 123===4AAArrr rAAAA        ,m where 123,,AAA and A are def i ned as Theorem 4. Corollary 2.10 Suppose 01 23=.AAAiAjAk  Then a) A quaternion matrix A has a pure imaginary mini-mum norm g-inverse if and only if 0=.ArrAA ,AAAAAAAAAAAAAAAAAAAA b) All minimum norm g-inverse of quaternion matrix A are pure imaginary matrices if and only if 0=4,ArrAA m where 0A and A are define d as Theorem 4. Copyright © 2011 SciRes. ALAMT H. S. ZHANG Copyright © 2011 SciRes. ALAMT 7REFERENCES  T. W. Hungerford, “Algebra,” Spring-Verlag Inc., New York, 1980.  A. Ben-Israel and T. N. E. Greville, “Generalized Inver- ses: Theory and Applications,” R. E. 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