Algebraic proof methods for identities of matrices and operators: improvements of Hartwig's triple reverse order law
Dragana S. Cvetković-Ilić, Clemens Hofstadler, Jamal Hossein Poor, Jovana Milošević, Clemens G. Raab, Georg Regensburger
aa r X i v : . [ m a t h . F A ] A ug Algebraic proof methods for identities of matrices andoperators: improvements of Hartwig’s triple reverseorder law
Dragana S. Cvetkovi´c-Ili´c , Clemens Hofstadler , Jamal Hossein Poor ,Jovana Miloˇsevi´c , Clemens G. Raab , and Georg Regensburger Department of Mathematics, Faculty of Sciences and Mathematics,University of Niˇs, Serbia Institute for Algebra, Johannes Kepler University Linz, Austria
Abstract
When improving results about generalized inverses, the aim often is to do thisin the most general setting possible by eliminating superfluous assumptions and bysimplifying some of the conditions in statements. In this paper, we use Hartwig’swell-known triple reverse order law as an example for showing how this can be doneusing a recent framework for algebraic proofs and the software package
OperatorGB .Our improvements of Hartwig’s result are proven in rings with involution and wediscuss computer-assisted proofs that show these results in other settings based onthe framework and a single computation with noncommutative polynomials.
Keywords : matrices and linear operators, algebraic operator identities, generalized in-verses, reverse order law, automated proofs, noncommutative polynomials, quiver repre-sentations
MSC 2020 : 15A09, 68V15, 03B35 (Primary); 16B50, 16G20 (Secondary)
Introducing generalized inverses and developing tools working with them in the casewhen ordinary inverses do not exist, resulted in a lot of progress in several branches ofmathematics and many other fields outside of mathematics (mechanics, robotics, con-trol theory, automation, etc.). The importance and usefulness of this area of researchis demonstrated by various open problems that have been solved using the theory ofgeneralized inverses and by many published results. However, a lot of recently publishedresults for generalized inverses and their applications were proved only under restrictiveassumptions which limit their applications to certain very particular cases. One reasonfor that is that, in contrast to the setting of matrices, generalized inverses are not definedfor each element of more general settings considered (algebras of operators, C ∗ -algebras,rings, . . . ). In order to benefit from the rich theory of generalized inverses and many1lready developed useful techniques, researchers usually impose existence of generalizedinverses when proving statements. This leads to many results with redundant instancesof assuming regularity of certain elements which makes them less applicable.The basic example for unnecessary regularity assumptions is the matrix equation AXB = C , which was one of the first applications of the later called Moore-Penrose inverse thatwas introduced by Moore and Penrose independently. Its solvability and the generalsolution were considered by Penrose in 1955 [1] in the same paper in which he introducedthe four Penrose equations. Since this result is almost algebraic, it was very easy togeneralize it for example to the case of operator equations AXB = C but under theadditional assumptions of the closedness of the ranges of the bounded linear operators A and B (that is equivalent with the existence of their Moore-Penrose inverses for operatorson Hilbert spaces). Solvability of this equation in the general case was only consideredseveral years ago, see [2], but many other problems, such as, for example, the existence ofa positive solution of that same equation, are still open in the general case. In fact, thereare a lot of problems like this where we have an answer only in some particular cases.So, in the recent years a lot of effort has been made to widen the range of applicabilityof these results by considering more general cases of the problems without imposing anyadditional assumptions. This paper is exactly one such important step in generalizingHartwig’s triple reverse order law.In this paper, we present several significant improvements of Hartwig’s triple reverseorder law motivated by using the software package OperatorGB [3], which is based on[4, 5]. The aim is to prove statements in an abstract setting in such a way that anal-ogous statements in various concrete settings (e.g. for matrices, linear bounded opera-tors, C ∗ -algebras, . . . ) can easily be proven in a rigorous way, but without inspectingevery step of the proof of the abstract statement. To this end, we employ a recentframework that allows to produce rigorous proofs for several different concrete settingsby translating a single statement about abstract noncommutative polynomials. Thisframework was developed in [4] and the software package OperatorGB provides extensivecomputer support for doing the computations needed. In particular, the software pro-vides explicit certificates of identities, which can be checked independently. Moreover,the software can also be used to explore variations of given statements. That is whatinitiated the improvements of Hartwig’s triple reverse order law presented in this paper.Based on the results obtained by this software we give a hand proof in the setting ofrings which hopefully provides motivation for further research with the same idea. Inaddition, we explain how computer-assisted proofs of all these improvements can be doneand we provide a
Mathematica notebook containing all these automated proofs at http://gregensburger.com/softw/OperatorGB . These improvements are the first newresults that are obtained by applying the framework and software. From this websitealso a
Mathematica as well as a
SageMath version of the
OperatorGB package canbe obtained.The main setting that we consider in this paper is a ring R with a unit 1 = 0 and aninvolution a a ∗ satisfying( a ∗ ) ∗ = a, ( a + b ) ∗ = a ∗ + b ∗ , ( ab ) ∗ = b ∗ a ∗ . Definition 1.1.
We say that a ∈ R is Moore-Penrose invertible (or MP-invertible) , ifthere exists b ∈ R such that the following hold: aba = a, bab = b, ( ab ) ∗ = ab, ( ba ) ∗ = ba. (1)2 n element b that satisfies (1) is called a Moore-Penrose inverse of a . It is well known that the Moore-Penrose inverse is unique when it exists. We denote theMoore-Penrose inverse of a by a † . We point out some properties of the Moore-Penroseinverse that follow from the definition. Clearly, a is MP-invertible if and only if a ∗ isMP-invertible; in this case ( a ∗ ) † = ( a † ) ∗ . If a is MP-invertible, then so are a ∗ a and aa ∗ , with( a ∗ a ) † = a † ( a ∗ ) † , ( aa ∗ ) † = ( a ∗ ) † a † . Definition 1.2.
An element a ∈ R is left ∗ -cancellable if, for all z ∈ R , a ∗ az = 0 implies az = 0 , it is right ∗ -cancellable if, for all z ∈ R , zaa ∗ = 0 implies za = 0 , and ∗ -cancellable if it is both left and right cancellable. We observe that a is left ∗ -cancellable if and only if a ∗ is right ∗ -cancellable. In a C ∗ -algebra, every element is ∗ -cancellable: If a ∗ az = 0, then ( az ) ∗ az = 0 which implies az = 0; similarly zaa ∗ = 0 implies za = 0.If b ∈ R satisfies { i, . . . , j } of the Penrose equations from (1) we say that b is a { i, . . . , j } -inverse of a . The set of all { i, . . . , j } -inverses of a is denoted by a { i, . . . , j } . Evidently a { , , , } = { a † } . We say that an element a ∈ R is regular if a { } 6 = ∅ . In general,in C ∗ -algebras we have that the regularity property is equivalent with MP-invertibility.In particular, in an algebra of bounded linear operators the regularity of an arbitraryoperator A is equivalent to the closedness of the range of A while in a ring with involutionMP-invertibility of m is equivalent to the right ∗ -cancellability of m and group invertibilityof mm ∗ (see Theorem 8.25 from [6] or Theorem 5.3 from [7]). Definition 1.3.
An element a ∈ R is EP if a R = a ∗ R . In the following subsection, we give a self-contained informal overview of the frameworkfor algebraic proofs and of the software package
OperatorGB . In Section 2, we first dis-cuss Hartwig’s triple reverse order law and related results from the literature. Then, wegive hand proofs of several improvements of it in rings with involution. After that, inSection 2.1, we discuss how these results can be proven with the help of the computerin such a way that the framework yields rigorous proofs for these statements also in thecontext of matrices and operators. Formal definitions and statements about the frame-work for algebraic proofs, which is used by the software
OperatorGB , are summarized inthe appendix.
The advantage of the framework presented below is that a single computation in an ab-stract setting proves analogous statements in various concrete settings (e.g. for matrices,linear bounded operators, C ∗ -algebras, . . . ) without having to inspect every step of theabstract computation. Just like in any ring, computations with noncommutative poly-nomials allow any two elements to be added or multiplied. Therefore, it is not clear apriori that a given proof of a statement in a ring is valid also for rectangular matrices or3perators with domains and codomains. Using the framework for algebraic proofs, thefollowing steps have to be carried out once in a suitable ring of noncommutative poly-nomials. Then, to rigorously prove a statement for various concrete settings, based onTheorem A.1, it suffices to check that the polynomials corresponding to the assumptionsand claims are compatible with different domains and codomains of operators.1. Express all assumptions and claimed properties as identities in terms of operators.2. Take the differences of the left and right hand sides of these identities and replacethe individual operators uniformly by noncommutative indeterminates in order toconvert the identities into polynomials.3. Find a concrete representation of the polynomials corresponding to the claim asa two-sided linear combination of polynomials corresponding to the assumptions,where coefficients are polynomials.Representations of polynomials as mentioned in the last step are called cofactor represen-tations and serve as certificates for ideal membership that can be checked independentlyof how they were found. However, finding them is a hard problem, since for noncommu-tative polynomials ideal membership is undecidable in general, see e.g. [8]. In practice,cofactor representations often can be found by computing a (partial) Gr¨obner basis, see[5] and references therein. Already in the pioneering work [9, 10] Gr¨obner bases have beenused to simplify matrix identities in linear systems theory. Proving operator identitiesusing Gr¨obner basis computations and related questions are also addressed in [11].The software package OperatorGB provides the command
Certify , which not only triesto compute cofactor representations but also does the compatibility checks of assumptionsand claims. Inspecting the explicit cofactor representations found by the software canalso give hints how assumptions could be relaxed by dropping the assumptions that donot appear in the cofactor representations. More generally, the software makes it easy toexperiment with different sets of assumptions for proving a desired claim. Improvementsof Hartwig’s triple reverse order law found by such experiments were the basis for theresults presented in the next section. For details on how our framework and software areused to find and prove these results, see Section 2.1.Next, we illustrate the approach with a simple statement about inner inverses of matrices,for details of the framework see the appendix. In [12, Thm. 2.3], Werner proved amongother things the following statement about inner inverses of complex matrices. If A and B are complex matrices such that AB exists, then N ( A ) ⊆ R ( B ) implies that B { } A { } ⊆ ( AB ) { } . As a first step, we have to phrase all properties stated in theassumptions and in the claim in terms of identities of matrices, which results in thefollowing statement. For any complex matrices A − , B − with AA − A = A and BB − B = B, (2)we have that BB − ( I − A − A ) = I − A − A (3)implies ABB − A − AB = AB. (4)The formats of these matrices can be visualized by the following diagram.4 m C n C k A − A I B − B Secondly, we represent these identities by noncommutative polynomials in the indeter-minates { a, a − , b, b − , i } . This is done by uniformly replacing each matrix (including theidentity matrix) by an indeterminate and forming the difference of the left and right handside of each identity. f = aa − a − a f = bb − b − b f = bb − ( i − a − a ) − i + a − a (5) f = abb − a − ab − ab (6)Moreover, for correctly handling the identity matrix, we also need to represent its alge-braic identities in terms of polynomials. f = ai − a f = ia − − a − f = ib − b f = b − i − b − f = i − i (7)Finally, either by hand or with the help of software, we can express the polynomial f representing the claim in terms of the polynomials f , . . . , f representing the assumptions. f = f b + af − af b + ( abb − − a ) f (8)By Theorem A.1, it follows from (8) that (4) holds for any matrices A, B with innerinverses A − , B − satisfying (3), see Lemma A.2 in the appendix. Moreover, based on thetheorem, the cofactor representation (8) also proves the analogous statement for boundedlinear operators A, B between Hilbert spaces
U, V, W as in the following diagram.
W V UA − A id B − B As mentioned above, explicit cofactor representations not only certify ideal membership,but can also give hints how assumptions could be relaxed. In particular, they also allowto analyze which assumptions can be relaxed for proving a given identity of operators.For example, (8) does not involve f , f , f , f , so in BB − ( I − A − A ) = I − A − A we couldreplace the identity matrix I by any other matrix J satisfying J B = B . Trivially, anycofactor representation with polynomials having only integer coefficients, as in (8) above,also holds in any ring, and hence proves an analogous statement for rings.As discussed before, to apply the proof framework directly, one has to translate all prop-erties of the operators involved into identities. In the context of generalized inverses,such properties are often conditions on ranges and kernels of some basic operators. Ifa projection (idempotent) on these spaces can be expressed in terms of basic operators,the translation to identities is immediate, as illustrated in the example above. Inclu-sion of ranges R ( A ) ⊆ R ( B ) can be translated in many situations to the existence ofa factorization A = BC for some operator C . In Hilbert or Banach spaces, this is thewell-known factorization property in Douglas’ lemma. For proving the existence of sucha linear operator C without any additional properties, one just needs operators definedon a vector space over an arbitrary field. This principle will play a prominent role inSection 2.1. 5 Improvements of Hartwig’s triple reverse order law
The “reverse order law” problem was originally posed by Greville [13] as early as in the1960’s, who first considered it in the case of the Moore-Penrose inverse of the product oftwo matrices. Namely, for given matrices
A, B such that AB is defined the following wasproved: ( AB ) † = B † A † ⇔ R ( A ∗ AB ) ⊆ R ( B ) , R ( BB ∗ A ∗ ) ⊆ R ( A ∗ ) . (9)This was followed by further research on this subject branching in several directions:- for products of more than two matrices,- for different classes of generalized inverses ( { } , { , } , { , , } , etc.), and- in different settings (operator algebras, C ∗ -algebras, rings, etc.).For more information on this subject please see [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,25, 26, 27, 28, 29, 30, 31, 32].One of the first to be inspired by Greville’s result (9) was Hartwig [33], who studied thereverse order law for the Moore-Penrose inverse of the product of three matrices. Indeed,he considered necessary and sufficient conditions such that( ABC ) † = C † B † A † (10)holds. Theorem 2.1. [ ] Let
A, B, C be complex matrices such that
ABC is defined and let P = A † ABCC † , Q = CC † B † A † A . The following conditions are equivalent: ( i ) ( ABC ) † = C † B † A † ;( ii ) Q ∈ P { , } and both of A ∗ AP Q and
QP CC ∗ are Hermitian; ( iii ) Q ∈ P { , } and both of A ∗ AP Q and
QP CC ∗ are EP; ( iv ) Q ∈ P { } , R ( A ∗ AP ) = R ( Q ∗ ) and R ( CC ∗ P ∗ ) = R ( Q ) ; ( v ) P Q = (
P Q ) , R ( A ∗ AP ) = R ( Q ∗ ) and R ( CC ∗ P ∗ ) = R ( Q ) . This inspired many authors to continue research in these directions and it was preciselyHartwig’s result that motivated further consideration of the reverse order law for MP-inverses in the case of three elements in certain other settings such as in the algebra ofbonded linear operators and in C ∗ -algebras, which was done in [34] and [35], respectively.In both papers, results analogous to Hartwig’s paper were obtained, but with the addi-tional conditions of regularity of all three elements and their products. Here, we mentiona result presented in [35] for the case of C ∗ -algebras in order to give a clear picture ofthe conditions assumed and the equivalences obtained (in the case of bounded linearoperators between Hilbert spaces the theorem looks identically).6 heorem 2.2. [ ] Let A be a complex unital C ∗ -algebra and let a, b, c ∈ A be suchthat a, b, c and abc are regular. Let p = a † abcc † and q = cc † b † a † a . Then, the followingconditions are equivalent: ( i ) ( abc ) † = c † b † a † ;( ii ) q ∈ p { , } and both of a ∗ apq and qpcc ∗ are Hermitian; ( iii ) q ∈ p { , } and both of a ∗ apq and qpcc ∗ are EP; ( iv ) q ∈ p { } , a ∗ ap A = q ∗ A and cc ∗ p ∗ A = q A ; ( v ) pq = ( pq ) , a ∗ ap A = q ∗ A and cc ∗ p ∗ A = q A . The main results presented here represent an important improvement of Hartwig’s resultin several senses: ◦ We consider the problem in rings with involution, which is a more abstract settingthan what was considered in the literature so far. Together with the framework andthe discussion in Section 2.1 this generalizes all the results previously mentioned. ◦ We relax conditions ( iv ) and ( v ) in the original result of Hartwig (Theorem 2.1),by replacing the respective equalities of ranges assumed in both of these conditionswith appropriate inclusions of ranges. For example, we show in Theorems 2.3 and2.4 that certain combinations of inclusions (there are four of them in total), alongwith the assumption that the element pq is idempotent, imply (10), while the othertwo combinations do not guarantee the claimed conclusion (see Example 2.5). Asfor the analogous results for algebras of operators and C ∗ -algebras (see [34] and[35]), we improve them in a similar way by replacing equalities with appropriateinclusions. ◦ Compared to the results for algebras of operators and C ∗ -algebras in general (see[34] and [35]), we significantly reduce the set of starting assumptions upon whichthese results are based by dropping certain regularity conditions. Namely, if oneis interested in the validity of (10), it is possible to omit the requirement that theproduct abc is MP-invertible, since this follows directly from some of the assump-tions ( iv ) or ( v ). In the case of rings, MP-invertibility of the product abc can bereplaced with the weaker condition of right ∗ -cancellability of abc . See Theorems 2.3and 2.6 and similarly Theorem 2.4. ◦ Also, it is possible to generalize the result by showing that b † can be replaced by anarbitrary element e b as well as that a † and c † can be replaced with arbitrary a (1 , , and c (1 , , , respectively (see Theorem 2.7). In this way, the assumption of MP-invertibility of the element b is dropped and the MP-invertibility of the elements a and c is replaced with the existence of a (1 , , and c (1 , , . This, although the last twoare equivalent conditions in operator algebras and C ∗ -algebras, improves the resultssignificantly in rings with involution since there the existence of a { , , } -inverseof an element is equivalent with the existence of its { , } -inverse and the latter is amuch weaker condition than MP-invertibility (as witnessed by the ring M ( C ) withtaking transposes as the involution). 7ecall that R denotes a ring with a unit 1 = 0 and with an involution. Theorem 2.3.
Let a, b, c ∈ R be such that a, c are MP-invertible. Let p = a † abcc † and q = cc † e ba † a , for e b ∈ R . Then, the following conditions are equivalent: ( i ) abc is Moore-Penrose invertible and ( abc ) † = c † e ba † ; ( iv ) q ∈ p { } , a ∗ ap R ⊇ q ∗ R and cc ∗ p ∗ R ⊆ q R ; ( v ) abc is right ∗ -cancellable, pq = ( pq ) , a ∗ ap R ⊇ q ∗ R and cc ∗ p ∗ R ⊆ q R ; ( vi ) q ∈ p { } , a ∗ ap R ⊇ q ∗ R and cc ∗ p ∗ R ⊆ q R . Proof.
Let m = abc and e m = c † e ba † . Evidently, pq is idempotent if and only if m e m isidempotent. Also, we have that the following equivalences hold: a ∗ ap R ⊇ q ∗ R ⇔ m R ⊇ ( e m ) ∗ R ⇔ R m ∗ ⊇ R e m ⇔ e m ∈ R m ∗ ; cc ∗ p ∗ R ⊆ q R ⇔ m ∗ R ⊆ e m R ⇔ m ∗ ∈ e m R ;( i ) ⇒ ( v ): If m † = e m , then clearly m e m is idempotent. Also, e m = m † = m † mm † = m † ( m † ) ∗ m ∗ ∈ R m ∗ ,m ∗ = ( mm † m ) ∗ = m † mm ∗ = e mmm ∗ ∈ e m R . ( v ) ⇒ ( i ): If ( v ) holds, then there exist u, v ∈ R such that e m = um ∗ and m ∗ = e mv .Now, multiplying m e m = ( m e m ) by v from the right side, we get mm ∗ = m e mmm ∗ i.e.(1 − m e m ) mm ∗ = 0, which gives (1 − m e m ) m = 0 by right ∗ -cancellability of m . So, e m isan inner inverse of m . Further, we have that e m = um ∗ = u ( m e mm ) ∗ = e m ( m e m ) ∗ , which implies that m e m is Hermitian and further e m = e m ( m e m ) ∗ = e mm e m. Also, m = v ∗ ( e m ) ∗ = v ∗ ( e mm e m ) ∗ = m ( e mm ) ∗ , which implies that e mm is Hermitian.( iv ) , ( vi ) ⇒ ( v ): This is evident.( i ) ⇒ ( iv ): The property q ∈ p { } follows directly from the fact that e m is an innerinverse of m . The rest of the proof follows as in the part ( i ) ⇒ ( v ).( i ) ⇒ ( vi ): The property q ∈ p { } follows from the fact that e m is an outer inverse of m .The rest of the proof follows as in the part ( i ) ⇒ ( v ). (cid:3) It is interesting to mention that if we take the reverse inclusion from ( ii ) of Theorem2.3 (notice that in Hartwig’s result we have equality!) and replace in the statement ofthe theorem the assumption of right ∗ -cancellability of abc with the assumption of left ∗ -cancellability of c † e ba † , we get the following analogous result.8 heorem 2.4. Let a, b, c, e b ∈ R be such that a, c are MP-invertible. Let p = a † abcc † and q = cc † e ba † a . Then, the following conditions are equivalent: ( i ) abc is Moore-Penrose invertible and ( abc ) † = c † e ba † ; ( iv ) q ∈ p { } , a ∗ ap R ⊆ q ∗ R and cc ∗ p ∗ R ⊇ q R ; ( v ) c † e ba † is left ∗ -cancellable, pq = ( pq ) , a ∗ ap R ⊆ q ∗ R and cc ∗ p ∗ R ⊇ q R ; ( vi ) q ∈ p { } , a ∗ ap R ⊆ q ∗ R and cc ∗ p ∗ R ⊇ q R . The following example illustrates the fact that the remaining two combinations of in-clusions in the original result of Hartwig (Theorem 2.1 ( v )) do not necessarily imply(10). Example 2.5.
Let A = − , B = , C = 13 . Then A † = 117 − , B † = − − , C † = C. If we define P and Q as in Theorem 2.1, we get that P Q = 0 is idempotent and R ( A ∗ AP ) ⊆ R ( Q ∗ ) and R ( CC ∗ P ∗ ) ⊆ R ( Q ) but ( ABC ) † = C † B † A † .If matrices A, B, C are defined as C † , B † and A † , respectively, as given above, we concludethat also the second pair of inclusions R ( Q ∗ ) ⊆ R ( A ∗ AP ) and R ( Q ) ⊆ R ( CC ∗ P ∗ ) together with the assumption that the matrix P Q is idempotent fails to imply (10) . On the other hand, the above mentioned pairs of inclusions imply (10) with some as-sumptions on p and q . Theorem 2.6.
Let a, b, c ∈ R be such that a, c are MP-invertible. Let p = a † abcc † and q = cc † e ba † a , for e b ∈ R . Then, the following conditions are equivalent: ( i ) abc is Moore-Penrose invertible and ( abc ) † = c † e ba † ; ( iv ) q ∈ p { } , a ∗ ap R ⊇ q ∗ R and cc ∗ p ∗ R ⊇ q R ; ( vi ) q ∈ p { } , a ∗ ap R ⊆ q ∗ R and cc ∗ p ∗ R ⊆ q R . In addition to the previously mentioned results, we can show that MP-invertibility of theelements a and c can be replaced with the existence of a (1 , , and c (1 , , . Theorem 2.7.
Let a, b, c, e b ∈ R be such that there exist a (1 , and c (1 , and such that abc is right ∗ -cancellable. Let a (1 , , , c (1 , , be given such that c (1 , , e ba (1 , , is left ∗ -cancellableand let p = a (1 , , abcc (1 , , and q = cc (1 , , e ba (1 , , a . Then, the following conditions areequivalent: i ) abc is Moore-Penrose invertible and ( abc ) † = c (1 , , e ba (1 , , ; ( ii ) q ∈ p { , } and both of a ∗ apq and qpcc ∗ are Hermitian; ( iii ) q ∈ p { , } and both of a ∗ apq and qpcc ∗ are EP; ( iv ) pq = ( pq ) , a ∗ ap R ⊇ q ∗ R and cc ∗ p ∗ R ⊆ q R ; ( v ) pq = ( pq ) , a ∗ ap R ⊆ q ∗ R and cc ∗ p ∗ R ⊇ q R . Notice that, if in Theorem 2.7 we replace a (1 , , and c (1 , , with a (1 , and c (1 , , respec-tively, the assertion of the theorem does not hold anymore, which will be shown in thenext example: Example 2.8.
Let B = C = e B = I and take any matrix A such that A { , , } 6 = { A † } (such A can be any projection different from the identity). If we take A (1 , = A (1 , , = A † we get that the conditions ( ii ) − ( v ) are all satisfied while ( i ) from Theorem 2.7 is notsatisfied. Finally, by the discussion above we end this section with the improved version of Hartwig’soriginal result for matrices.
Theorem 2.9.
Let
A, B, C be complex matrices such that
ABC is defined and let P = A † ABCC † , Q = CC † B † A † A . The following conditions are equivalent: ( i ) ( ABC ) † = C † B † A † ;( ii ) Q ∈ P { , } and both of A ∗ AP Q and
QP CC ∗ are Hermitian; ( iii ) Q ∈ P { , } and both of A ∗ AP Q and
QP CC ∗ are EP; ( iv ′ ) Q ∈ P { } , R ( Q ∗ ) ⊆ R ( A ∗ AP ) and R ( CC ∗ P ∗ ) ⊆ R ( Q ) ; ( iv ′′ ) Q ∈ P { } , R ( A ∗ AP ) ⊆ R ( Q ∗ ) and R ( Q ) ⊆ R ( CC ∗ P ∗ ) ; ( iv ′′′ ) Q ∈ P { } , R ( Q ∗ ) ⊆ R ( A ∗ AP ) and R ( Q ) ⊆ R ( CC ∗ P ∗ ) ; ( iv ′′′′ ) Q ∈ P { } , R ( Q ∗ ) ⊆ R ( A ∗ AP ) and R ( CC ∗ P ∗ ) ⊆ R ( Q ) ; ( iv ′′′′′ ) Q ∈ P { } , R ( A ∗ AP ) ⊆ R ( Q ∗ ) and R ( Q ) ⊆ R ( CC ∗ P ∗ ) ; ( iv ′′′′′′ ) Q ∈ P { } , R ( A ∗ AP ) ⊆ R ( Q ∗ ) and R ( CC ∗ P ∗ ) ⊆ R ( Q ) ; ( v ′ ) P Q = (
P Q ) , R ( Q ∗ ) ⊆ R ( A ∗ AP ) and R ( CC ∗ P ∗ ) ⊆ R ( Q ) ; ( v ′′ ) P Q = (
P Q ) , R ( A ∗ AP ) ⊆ R ( Q ∗ ) and R ( Q ) ⊆ R ( CC ∗ P ∗ ) . .1 Computer-assisted algebraic proofs In the following, we discuss different aspects and use cases of the proof framework out-lined in Section 1.1. We use Hartwig’s result and its improvements presented aboveto exemplify this. Algebraically, the central point of the proof is membership of thepolynomial representing the claimed identity in the ideal generated by the polynomialsrepresenting the assumed identities, c.f. the third step listed in the introduction. Be-low, we also describe how certain assumptions, which are not identities of matrices oroperators themselves, can sometimes still be used within the framework.First, we focus on the implication ( v ) ⇒ ( i ) in Theorem 2.1: if P QP Q = P Q , R ( A ∗ AP ) = R ( Q ∗ ), and R ( CC ∗ P ∗ ) = R ( Q ), then M † = C † B † A † .Based on Douglas’ lemma, we first translate the range conditions to identities of operators.The four inclusions of ranges are equivalent to the following identities for some operators U , U , V , V . A ∗ AP = Q ∗ V A ∗ AP V = Q ∗ CC ∗ P ∗ = QU CC ∗ P ∗ U = Q (11)For each Moore-Penrose inverse A † , B † , C † , M † , we have the four defining identities.Translating these identities into polynomials, we introduce an indeterminate for eachbasic operator. Moreover, for each indeterminate, we introduce another indeterminaterepresenting the adjoint of the corresponding operator. In total, this amounts to 22indeterminates. Similarly, each identity of operators is translated into two polynomials,one for the identity itself and one for its adjoint. Thereby, we obtain a set F of 34noncommutative polynomials with integer coefficients representing the assumptions. Theclaim corresponds to the polynomial f = m † − c † b † a † .Then, we use our software to show that f lies in the ideal generated by the polynomi-als of F . The cofactor representation certifying this ideal membership was computedin less than 45 seconds and has 157 terms. The diagram induced by generic domainsand codomains of operators has 4 vertices and one edge for each indeterminate. Byconstruction, the polynomial f and the elements of F are compatible with domains andcodomains. By Theorem A.1, this now rigorously proves that M † = C † B † A † holds underthe conditions given in ( v ). Note that this proof only relies on the defining identitiesof Moore-Penrose inverses and does not use any additional properties or lemmas. Con-sequently, the implication ( v ) ⇒ ( i ) is in fact proven for any setting in which it canbe formulated, since the polynomials in the cofactor representation obtained have onlyinteger coefficients.Using the software, it is easy to experiment with relaxing the assumptions and check ifa cofactor representation of f in terms of a subset of F still can be found. For instance,it turns out that the first and last identity in (11) can be dropped. This correspondsto relaxing the range conditions in ( v ) to R ( A ∗ AP ) ⊇ R ( Q ∗ ) and R ( CC ∗ P ∗ ) ⊆ R ( Q ).Additionally, we could also observe that the cofactor representation of f contains nopolynomial associated to any of the four defining equations of B † . This shows that B † can in fact be replaced by an arbitrary operator ˜ B that does not have to be related to B in any way.It is also possible to prove the implication ( i ) ⇒ ( v ) using our framework and software.To this end, first explicit expressions for U , U , V , V in terms of the other basic operatorshave to be found. By inspecting the proof of Theorem 2.3 one can see that these can be11hosen as U = BCC ∗ B ∗ A ∗ ( A † ) ∗ , U = ( B † ) ∗ ( C † ) ∗ C † B † A † A,V = B ∗ A ∗ ABCC † , V = B † A † ( A † ) ∗ ( B † ) ∗ ( C † ) ∗ C ∗ . (12)Then, using the defining equations of A † , B † , C † , M † , the identity M † = C † B † A † andtheir adjoint statements as assumptions, the software finds cofactor representations ofthe polynomial corresponding to P QP Q = P Q as well as of the polynomials associatedto the four identities in (11), where U , U , V , V have been replaced by the expressions in(12). We note that these cofactor representations only contain polynomials with integercoefficients. Hence, based on Theorem A.1, this proves the implication ( i ) ⇒ ( v ) for anysetting in which it can be formulated.It is also possible to incorporate properties of operators into this framework that cannotbe expressed in terms of identities but only in form of quasi-identities. In general, quasi-identities are implications where a conjunction of identities implies another identity. Oneexample of such a property is ∗ -cancellability. To use these properties to prove a claimedidentity, first a suitable polynomial in the ideal representing the assumptions has to befound that corresponds to an operator identity to which such a property is applicable.Finding such a suitable polynomial is usually a non-trivial task and often has to bedone by hand. For the automated proofs of some of the results presented here, forexample, we obtained the required expressions by inspecting the corresponding handproofs, which were done partly before the automated proofs. Once such a polynomial hasbeen found, the corresponding quasi-identity can be applied to obtain a new polynomialthat corresponds to a shorter identity and that is typically not contained in the ideal thatis generated by the polynomials representing the assumptions. By including this newpolynomial into the set of polynomials representing the assumptions, we can enlarge theideal of all consequences of the assumptions and proceed to prove the ideal membershipof the polynomial corresponding to the claimed identity in this larger ideal.To prove a quasi-identity, the left-hand side of the implication has to be included inthe assumptions and the right-hand side becomes the claimed identity. When translatingthese operator identities into polynomials it is important to introduce new indeterminatesthat do not satisfy any additional identities for all universally quantified operators in thequasi-identity. Then, to prove the quasi-identity, it only remains to prove the idealmembership of the polynomial associated to the claim in the ideal generated by thepolynomials representing the assumptions.Based on the discussion and the observations made above, it is no surprise that thesoftware can also be used to prove all the improved results of Hartwig’s triple reverseorder law presented in this work. In the following, we explain how this can be done usingthe equivalence ( i ) ⇔ ( v ) of Theorem 2.3.For the implication ( v ) ⇒ ( i ), we translate the assumptions pq = ( pq ) , a ∗ ap R ⊇ q ∗ R , cc ∗ p ∗ R ⊆ q R and their adjoint statements into polynomials. Note that in order totranslate the set inclusions we can use factorizations analogous to (11). In contrast to theoriginal statement of Hartwig, where the MP-invertibility of ABC is already given, we nowhave to prove that m = abc is MP-invertible and that m † = c † ˜ ba † . Hence, the claim is that˜ m = c † ˜ ba † satisfies the four defining equations of m † . However, trying to show the idealmembership of the corresponding polynomials in the ideal generated by the polynomialsrepresenting the assumptions fails. This is because these polynomials do not contain any12nformation about the right ∗ -cancellability of m . To use this property, we have to find apolynomial in the ideal generated by the polynomials associated to our assumptions thatcorresponds to an identity to which this property is applicable. In the hand proof of thisimplication, the right ∗ -cancellability is applied to (1 − m ˜ m ) mm ∗ = 0. Using the software,we can show that the polynomial corresponding to this identity is indeed contained inthe ideal generated by the polynomials representing the assumptions. Hence, as in thehand proof, we can apply the right ∗ -cancellability of m to (1 − m ˜ m ) mm ∗ = 0 to obtain(1 − m ˜ m ) m = 0. After including the polynomial associated to this new identity in theset of translated assumptions, the software manages to verify the ideal membership ofall polynomials corresponding to the claimed identities fully automatically, and thereby,proves the claimed statement.The proof of ( i ) ⇒ ( v ) of Theorem 2.3 using the software essentially proceeds along thesame lines as the proof discussed above concerning the same implication in Hartwig’stheorem. The only difference is that now also the right ∗ -cancellability of m has tobe shown. To this end, we include the identity zmm ∗ = 0 in the assumptions andprove zm = 0 with an arbitrary ring element z . When translating these identities intopolynomials, z has to be replaced by a new indeterminate that does not satisfy anyadditional identities. The software then proves the ideal membership of the polynomialassociated to the claimed identity in the ideal generated by the polynomials representingthe assumptions fully automatically. Remark 2.10.
We note that in a similar fashion to the implications discussed above, alsoall other implications of Theorem 2.3 and all other results presented in this work, includingTheorems 2.1, 2.2, 2.3, 2.4, 2.6, 2.7, and 2.9, can be proven using the framework. Therelevant computations with noncommutative polynomials were done using
OperatorGB and are available at http: // gregensburger. com/ softw/ OperatorGB along with afile containing all the certificates of ideal membership. Since all cofactor representationsobtained have only polynomials with integer coefficients, by applying Theorem A.1, thecorresponding theorems hold for any setting in which they can be formulated like rings withinvolution, (rectangular) matrices over such rings, and linear bounded operators betweenHilbert spaces.
Acknowledgements
We thank Anja Korporal, Marko Petkovi´c, and Milan Tasi´c for discussions related to thispaper in the course of the OeAD project SRB 05/2016. This work was supported bythe Ministry of Science, Technology and Development, Republic of Serbia, and by theAustrian Science Fund (FWF): P 27229, P 31952, and P 32301.
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A Formal summary of algebraic proof framework
Now, we give a more formal explanation of the framework developed in [4]. In thefollowing, we fix a set X and a commutative ring R with unit element. We consider thering R h X i of noncommutative polynomials with coefficients in R and indeterminates in X , where indeterminates do not commute with each other but with coefficients.Recall that a quiver is given by a tuple ( V, E, s, t ) where V is the set of vertices, E is theset of edges, and s, t : E → V give the source s ( e ) and target t ( e ) of each edge e ∈ E .We consider labelled quivers where edges have labels in X , i.e. with a function l : E → X giving the labels of edges. In the following, we fix a labelled quiver Q = ( V, E, X, s, t, l )such that edges have unique labels, i.e. l is injective. Based on the labels of edges,it is straightforward to label paths in Q so that multiplication of labels as monomialscorresponds to concatenation of paths. Likewise, the notion of source and target of edgescan be naturally extended to paths.A polynomial in R h X i such that all its monomials are labels of paths in Q that have thesame source and the same target is called compatible with Q . For vertices v, w ∈ V , wecollect all compatible polynomials arising from paths with source v and target w in theset R h X i v,w , which is an R -module. Note that for the case v = w there exists an emptypath from v to w , which has the constant monomial 1 as its label. By construction, thepolynomials f , . . . , f , f defined in Section 1.1 are compatible with the following labelledquiver. • • • a − a i b − b Figure 1: Labelled quiver for Werner’s theoremA representation of a quiver (
V, E, s, t ) can be specified by a pair ( M , ϕ ) such that M = ( M v ) v ∈ V is a family of R -modules and the map ϕ assigns to each e ∈ E an R -linearmap ϕ ( e ) : M s ( e ) → M t ( e ) .For example, with R = Z or R = C , the two diagrams inSection 1.1 specify representations of the labelled quiver shown in Figure 1.Now, for a given representation ( M , ϕ ) of Q , plugging in the R -linear maps ϕ ( e ), e ∈ E ,for the indeterminates l ( e ) of polynomials in R h X i can be formalized as follows. Forevery nonconstant monomial m ∈ R h X i v,w , there exists a nonempty path e n . . .e in Q v , target w , and label m , which allows to define the R -linear map ϕ v,w ( m ) := ϕ ( e n ) · . . . · ϕ ( e ) from M v to M w . Note that, by definition of ϕ , the composition of themaps ϕ ( e i ) exists. Similarly, if v = w , we define ϕ v,v (1) := id M v . The map ϕ v,w extends R -linearly to all f ∈ R h X i v,w and we call the R -linear map ϕ v,w ( f ) a realization of f w.r.t. the representation ( M , ϕ ) of Q .Altogether, one can prove the following main theorem about the framework. The formu-lation stated here is a consequence of Theorem 32 and 15 in [4]. Theorem A.1.
Let R be a commutative ring with unit element, let F ⊆ R h X i be a setof polynomials without a constant term, and let f ∈ ( F ) . Then, for all labelled quivers Q with unique labels in X such that f and all polynomials in F are compatible with Q and for all representations ( M , ϕ ) of Q such that the realizations of the polynomials in F w.r.t. ( M , ϕ ) are zero, we have that also the realization of f w.r.t. ( M , ϕ ) is zero. All notions and results of this section naturally generalize to R -linear categories by consid-ering objects and morphisms in such a category instead of R -modules and R -linear maps,respectively. For more details, see Section 5.2 in [4]. Based on a refined version of theframework using rewriting, it is possible to obtain a similar theorem where polynomialsin F are allowed to have a constant term, see Theorem 32 in [36].Altogether, based on the theorem above, we obtain a rigorous proof of the followingstatement for matrices discussed in Section 1.1. Lemma A.2.
Let