In this paper, we study representations of the Moore-Penrose inverse of a 2 x 2  matrix M over a *-regular ring with two term star-cancellation.
As applications, some necessary and sufficient conditions for the Moore-Penrose inverse of M to have different types are given.This research is supported by the National Natural Science Foundation of China (11201063) and (11371089), the Specialized Research Fund for the Doctoral Program of Higher Education (20120092110020), the Foundation of Graduate Innovation Program of Jiangsu Province(CXLX13-072) and the Fundamental Research Funds for the Central Universities (22420135011), `FEDER Funds through "Programa Operacional Factores de Competitividade-COMPETE' and the Portuguese Funds through FCT-`Fundação para a Ciência e a Tecnologia', within the project PEst-OE/MAT/UI0013/2014

Chen, Jianlong

Patrício, Pedro

Zhang, Xiaoxiang

Zhu, Huihui

Applied Mathematics and Computation

Universidade do Minho: RepositoriUM

The Moore-Penrose inverse of 2 x 2 matrices over a certain *-regular ring

The Moore-Penrose inverse of 2× 2 matrices over acertain ∗-regular ringHuihui Zhua, Jianlong Chena,∗, Xiaoxiang Zhanga, Pedro Patr´ıciobaDepartment of Mathematics, Southeast University, Nanjing 210096, China.bCMAT-Centro de Matema´tica and Departamento de Matema´tica e Aplicac¸o˜es,Universidade do Minho, Braga 4710-057, Portugal.AbstractIn this paper, we study representations of the Moore-Penrose inverse of a2 × 2 matrix M over a ∗-regular ring with two term star-cancellation. Asapplications, some necessary and sufficient conditions for the Moore-Penroseinverse of M to have different types are given.Keywords:Moore-Penrose inverse, ∗-regular ring, two term star-cancellation2010 MSC: 15A09, 16E50, 16W101. IntroductionRepresentations for the Moore-Penrose inverse (abbr. MP-inverse) ofmatrices over various settings attract wide interest from many scholars. Forinstance, Cline [1, 2] derived the representations for the MP-inverse of apartitioned complex matrix. Hung and Markham [7, 8] obtained the explicitformula for the MP-inverse of an m×n partitioned matrix. Recently, Hartwigand Patr´ıcio [6] obtained new expressions for the MP-inverse of the matrix[ a 0b d ]over a ∗-regular ring, extending some well known results for complexmatrices.This article is motivated by the papers [5, 6]. We investigate the MP-inverse of M = [ a cb d ] over a ∗-regular ring satisfying some additional con-∗Corresponding authorEmail addresses: ahzhh08@sina.com (Huihui Zhu), jlchen@seu.edu.cn (JianlongChen ), z990303@seu.edu.cn (Xiaoxiang Zhang), pedro@math.uminho.pt (PedroPatr´ıcio)Preprint submitted to Applied Mathematics and Computation January 2, 2015ditions. As applications, some necessary and sufficient conditions for thematrix M to have various types are obtained. Some results in [5, 6] aregeneralized.Let R be a unital ∗-ring, that is a ring with unity 1 and an involutiona 7→ a∗ satisfying (a∗)∗ = a, (ab)∗ = b∗a∗ and (a + b)∗ = a∗ + b∗. By Rm×nwe denote the set of m× n matrices over R. The involution on R induces amap Rm×n → Rn×m, (aij) 7→ (a∗ji) denoted still by ∗. A matrix A ∈ Rm×n issaid to have an MP-inverse if there exists B ∈ Rn×m such that the followingequations hold [10]:ABA = A, BAB = B, (AB)∗ = AB and (BA)∗ = BA.Any elementB ∈ Rn×m satisfying the equations above is called an MP-inverseof A. If such a B exists, it is unique and is denoted by A†.Following [4], a ∗-ring R is said to satisfy the k-term star-cancellation law(SCk) ifa∗1a1 + · · ·+ a∗kak = 0⇒ a1 = · · · = ak = 0for any a1, · · · , ak ∈ R. Note that a ∗-ring satisfying SC1 is known as a ∗-cancellable ring. A ring is said to be ∗-regular if it is regular and ∗-cancellable(see, e.g., [9]). It is well-known that R is a ∗-regular ring if and only if everyelement in R is MP-invertible, and that R2×2 is a ∗-regular ring if and onlyif R is a regular ∗-ring satisfying SC2 (see, e.g., [6, p.182]).2. Main resultsThroughout this article we assume that R is a regular ∗-ring satisfyingSC2, an assumption that plays an essential role in Theorem 2.1 and Theorem2.7. (See Examples 2.2 and 2.8.). In particular, the rings R and R2×2 are∗-regular rings and every matrix M = [ a cb d ] in R2×2 is MP-invertible. Notethat M † = M∗(MM∗)† in this case (see [10, p.407]), a result that will bewidely-used in the sequel.If ab∗ + cd∗ = 0, as MM∗ =[aa∗+cc∗ ab∗+cd∗ba∗+dc∗ bb∗+dd∗]thenM † =[a∗(aa∗+cc∗)† b∗(bb∗+dd∗)†c∗(aa∗+cc∗)† d∗(bb∗+dd∗)†].Next theorem shows that the condition ab∗ + cd∗ = 0 is also necessary forsuch a decomposition to hold.2As usual, we denote the right annihilator of an element a in a ring R bya0. That is, a0 = {r ∈ R | ar = 0}.Theorem 2.1. Let R be a regular ∗-ring satisfying SC2 and M = [ a cb d ]∈ R2×2. Pose k = aa∗ + cc∗, l = bb∗ + dd∗ and m = ab∗ + cd∗. Then M † =[a∗k† b∗l†c∗k† d∗l†]if and only if m = 0.Proof. We need only to prove the “only if” part.First, we show that l0 ⊆ (b∗)0.Let x ∈ l0, i.e., (bb∗ + dd∗)x = 0. Then (b∗x)∗b∗x + (d∗x)∗d∗x = 0. SinceR satisfies SC2, we have b∗x = 0, i.e., x ∈ (b∗)0.Since 1− l†l ∈ l0, it follows that b∗ = b∗l†l and hence b = l∗(l∗)†b = ll†b.Similarly, d = ll†d.As[kk†a+ml†b kk†c+ml†dm∗k†a+ll†b m∗k†c+ll†d]= MM †M = M = [ a cb d ], one can seethat m∗k†a = 0 = m∗k†c, which implies m∗k†aa∗ = 0 = m∗k†cc∗. Hencem∗k†k = 0.Again, SC2 guarantees that k0 ⊆ (m∗)0 and hence m∗ = m∗k†k = 0.Consequently, m = 0.The next example shows that the assumption “R is a regular ∗-ring sat-isfying SC2” plays an essential role in Theorem 2.1.Example 2.2. Let R = Z/2Z with ∗ given by the identity map. Then R isa regular ∗-ring but it does not fulfil SC2 as 1∗1 + 1∗1 = 0 but 1 6= 0. Let M= [ a cb d ] = [1 11 1 ]. Then m = ab∗ + cd∗ = 0 but M † does not exist.Hartwig and Patr´ıcio [6] expressed the flipped MP-inverse of M = [ a 0b d ].Among others, they gave a necessary and sufficient condition under whichM † is of (2, 1, 0) type, i.e., the (2, 1) entry of M † is 0. Taking c = 0 inTheorem 2.1, we obtain a special case in which M † is of (2, 1, 0) type.Corollary 2.3. Let R be a regular ∗-ring satisfying SC2 and M = [ a 0b d ]∈ R2×2. Then M † =[a† b∗(bb∗+dd∗)†0 d∗(bb∗+dd∗)†]if and only if ab∗ = 0.3Theorem 2.4. A ring R is a regular ∗-ring satisfying SCn if and only ifevery n× 1 matrix[a1...an]over R is MP-invertible.Proof. “⇐” We first prove that R has the SCn property. Assume a∗1a1 +· · ·+a∗nan = 0 and α =[a1...an]. It follows thatA = [α, 0] ∈ Rn×n andA∗A = 0.As α† exists, A† =[α†0]. Note that A = (AA†)∗A = (A†)∗A∗A = 0. We seethat R has the SCn property.For a ∈ R, let[ a...0]†=[c1, · · · , cn]. Then c1 is the MP-inverse of a by adirect check.Therefore, R is a regular ∗-ring satisfying SCn.Conversely, let α =[a1...an]∈ Rn×1 and A = [α, 0] ∈ Rn×n. By hypothesisA† exists and set A† = β1...βn. It is easy to see α† = β1.Cline [2, Theorem 2] provided the presentation for the MP-inverse ofA+C, where A and C are complex matrices such that AC∗ = 0. His formulaindeed holds in the ring case, i.e., for any a, c ∈ R with ac∗ = 0,(a+ c)† = a† + (1− a†c)[u† + (1− u†u)vc∗(a†)∗a†(1− cu†)],where u = (1− aa†)c, w = a†c(1− u†u) and v = (1 +w∗w)−1. Note that theinvertibility of 1 +w∗w is guaranteed by our assumption at the beginning ofthis section (see [6, p. 182]).Hartwig and Patr´ıcio [6, p.183] simplified the above formula to(a+ c)† = (1 + y∗)(1 + yy∗)−1s+ u†,where u = (1−aa†)c, s = a†(1− cu†) and y = a†c(1−u†u). In addition, theyproved the following result.Lemma 2.5. [6, p.186] Let R be a regular ∗-ring satisfying SC2 and letA,C ∈ R2×2 with AC∗ = 0. If I + Y Y ∗ is invertible then(A+ C)† = (I + Y ∗)(I + Y Y ∗)−1S + U †,4where U = (I − AA†)C, S = A†(I − CU †) and Y = A†C(I − U †U).Lemma 2.6. Given a ∈ R, [ 1a ] is MP-invertible if and only if 1 + a∗a isinvertible.Proof. “⇒” Let [ 1a ]† = [b, c]. As ([ 1a ][b, c])∗ = [ 1a ][b, c], we have (ac)∗ = ac,b∗ = b and c∗ = ab. As [ 1a ] = [ 1a ] [b, c] [ 1a ], we get b+ ca = 1. So, (1 + a∗a)b =b∗ + a∗c∗ = (b+ ca)∗ = 1, and hence b∗(1 + a∗a) = 1.Conversely, pose y = (1 + a∗a)−1[1, a∗]. It is easy to check that y is theMP-inverse of [ 1a ].By virtue of Lemma 2.5, we can now prove our main theorem of thispaper. To calculate simply, we introduce the following notationse = a∗a+ b∗b, f = a∗c+ b∗d, g = c− ae†f, h = d− be†f,j = g∗g + h∗h, k = e†f(1− j†j), l = e†(a∗ − fj†g∗) and m = e†(b∗ − fj†h∗).Theorem 2.7. Let R be a regular ∗-ring satisfying SC2 and M = [ a cb d ]∈ R2×2.Then M † =[p rq s], wherep = (1 + kk∗)−1l, r = (1 + kk∗)−1m,q = j†g∗ + k∗(1 + kk∗)−1l and s = j†h∗ + k∗(1 + kk∗)−1m.Proof. Let A = [ a 0b 0 ], C = [0 c0 d ], U = (I − AA†)C, S = A†(I − CU †)and Y = A†C(I − U †U). As M = A + C and AC∗ = 0, then M † =(I + Y ∗)(I + Y Y ∗)−1S + U †.It is straightforward to check that A† = (A∗A)†A∗ =[e†a∗ e†b∗0 0]andU = (I − AA†)C =[0 g0 h]. Similarly, U † = (U∗U)†U∗ =[0 0j†g∗ j†h∗].HenceY = A†C(I − U †U) =[e†a∗ e†b∗0 0][ 0 c0 d ][1 00 1−j†j]= [ 0 k0 0 ]and5S = A†(I − CU †) =[e†a∗ e†b∗0 0] [1−cj†g∗ −cj†h∗−dj†g∗ 1−dj†h∗]= [ l m0 0 ].According to Theorem 2.4 and Lemma 2.6, it follows that 1 + kk∗ isinvertible and hence I + Y Y ∗ is invertible. Now, we have(I + Y ∗)(I + Y Y ∗)−1S = [ 1 0k∗ 1 ][(1+kk∗)−1 00 1][ l m0 0 ]=[(1+kk∗)−1l (1+kk∗)−1mk∗(1+kk∗)−1l k∗(1+kk∗)−1m].Therefore, the result follows by Lemma 2.5.The next example shows that the assumption “R is a regular ∗-ring sat-isfying SC2” is also essential for Theorem 2.7.Example 2.8. Let R = Z/2Z be as in Example 2.2. The following tableexhibits two cases in which M † = [ p rq s ] does not hold.TableM M † 1 + kk∗[p rq s][1 11 1]does not exist 1[0 00 0][1 01 1] [1 01 1]1[0 00 1]In the remainder of this section, we give some applications of Theorem2.7.Corollary 2.9. Under the hypothesis of Theorem 2.7, the following state-ments are equivalent:(1) M † =[(1+kk∗)−1e†a∗ (1+kk∗)−1e†b∗k∗(1+kk∗)−1e†a∗ k∗(1+kk∗)−1e†b∗].(2) j = 0.Proof. (2)⇒(1) is obvious.(1)⇒(2). As k∗(1 + kk∗)−1 = (1 + k∗k)−1k∗, thenM † =[(1+kk∗)−1e†a∗ (1+kk∗)−1e†b∗(1+k∗k)−1k∗e†a∗ (1+k∗k)−1k∗e†b∗].6Hence(1 + kk∗)−1e†a∗ = (1 + kk∗)−1[e†(a∗ − fj†g∗)] (2.1)and(1 + k∗k)−1k∗e†a∗ = j†g∗ + (1 + k∗k)−1k∗[e†(a∗ − fj†g∗)] (2.2)by Theorem 2.7. From (2.1) one can obtain e†fj†g∗ = 0. Combining thiswith (2.2), we get j†g∗ = 0.Similarly, it follows that j†h∗ = 0. Therefore, j = jj†j = jj†(g∗g + h∗h)= 0.A matrix M = [ a cb d ] with coefficients in R is said to be of (i, j, 0) typeif the (i, j) entry of M is zero. Note in [3, Corollary 2.7] that aa† = a†afor any a ∈ R† such that aa∗ = a∗a. It is easy to see that ee† = e†e sincee = a∗a+ b∗b.If M † is of (1, 1, 0) type, then p = 0 reduces to e†a∗=e†fj†g∗ and henceea∗ = efj†g∗. This implies ae = gj†f ∗e. We hence obtain the followingcorollary.Corollary 2.10. Let M = [ a cb d ]. Then M† is of (1, 1, 0) type if and only ifae = gj†f ∗e. In this case, we haveM † =[0 (1+kk∗)−1mj†g∗ j†h∗+k∗(1+kk∗)−1m].Corollary 2.11. Let M = [ a cb d ]. Then M† is of (1, 2, 0) type if and only ifbe = hj†f ∗e. In this case, we haveM † =[(1+kk∗)−1l 0j†g∗+k∗(1+kk∗)−1l j†h∗].IfM † is of (2, 1, 0) type, then q = j†g∗+k∗(1+kk∗)−1l = 0. By multiplyingthe above equations by 1 − j†j on the left, it follows that (1 − j†j)k∗(1 +kk∗)−1l = 0, that is k∗(1 + kk∗)−1l = 0. Hence k∗l = 0 since k∗(1 + kk∗)−1 =(1+k∗k)−1k∗. By substituting k∗l = 0 back into q, then follows that j†g∗ = 0.As (1 + kk∗)−1 = 1− (1 + kk∗)−1kk∗, we haveCorollary 2.12. Let M = [ a cb d ]. Then M† is of (2, 1, 0) type if and only ifj†g∗ = k∗l = 0. In this case, we haveM † =[l (1+kk∗)−1m0 j†h∗+k∗(1+kk∗)−1m].7Corollary 2.13. Let M = [ a cb d ]. Then M† is of (2, 2, 0) type if and only ifj†h∗ = k∗m = 0. In this case, we haveM † =[(1+kk∗)−1l mj†g∗+k∗(1+kk∗)−1l 0].ACKNOWLEDGMENTSThis research is supported by the National Natural Science Foundationof China (11201063) and (11371089), the Specialized Research Fund for theDoctoral Program of Higher Education (20120092110020), the Foundationof Graduate Innovation Program of Jiangsu Province(CXLX13-072) and theFundamental Research Funds for the Central Universities (22420135011),‘FEDER Funds through Programa Operacional Factores de Competitividade-COMPETE’ and the Portuguese Funds through FCT-‘Fundac¸a˜o para a Cieˆnciae a Tecnologia’, within the project PEst-OE/MAT/UI0013/2014. 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J. 25 (1975) 354-361.8[8] C.H. Hung, T.L. Markham, The Moore-Penrose inverse of a partitionedmatrix M = [ A DB C ], Linear Algebra Appl. 11 (1975) 73-86.[9] J.J. Koliha, P. Patr´ıcio, Elements of rings with equal spectral idempo-tents, J. Austral. Math. Soc. 72 (2002) 137-152.[10] R. Penrose, A generalized inverse for matrices, Proc. Camb. phil. Soc.51 (1955) 406-413.9

The Moore-Penrose inverse of 2 x 2 matrices over a certain *-regular ring

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Universidade do Minho: RepositoriUM