Representation of doubly infinite matrices as non-commutative Laurent series

Transcription

1 Spec. Matrices 217; 5: Research Article Open Access María Ivonne Arenas-Herrera and Luis Verde-Star* Representation of doubly infinite matrices as non-commutative Laurent series Received October 11, 217; accepted October 27, 217 Abstract: We present a new way to deal with doubly infinite lower Hessenberg matrices based on the representation of the matrices as the sum of their diagonal submatrices. We show that such representation is a simple and useful tool for computation purposes and also to obtain general properties of the matrices related with inversion, similarity, commutativity, and Pincherle derivatives. The diagonal representation allows us to consider the ring of doubly infinite lower Hessenberg matrices over a ring R as a ring of Laurent series in one indeterminate, with coefficients in the ring of R-valued sequences that don t commute with the indeterminate. Keywords: Doubly infinite matrices, non-commutative Laurent series, infinite Hessenberg matrices, similarity of infinite matrices, Pincherle derivatives. 1 Introduction Infinite matrices have been studied for a long time. They played a central role in the early stages of the development of Quantum Mechanics and Functional Analysis. The analytical problems that appear in the multiplication of infinite matrices, and the continuity of the operators represented by such matrices were extensively studied, but the study of the purely algebraic properties of the infinite matrices has not received the attention that it deserves. The spaces of triangular infinite matrices are the ones that have been studied in more detail and are often used in combinatorics, summability theory and some branches of algebra. On the other hand, the theory of doubly infinite matrices [a n, ], where the indices run over all the integers, is underdeveloped. Only some particular inds of bi-infinite matrices, such as bloc-toeplitz and Jacobi matrices, have been studied. Rings of convergence-free doubly infinite matrices were studied by Köthe and Toeplitz in [4]. Jabotiny [3] represented composition operators by means of doubly infinite matrices to study iteration of analytic functions. Barnabei, Brini, and Nicoletti studied recurrence properties and connections with the Umbral Calculus of doubly infinite matrices in [1]. More recently, several authors have extended the Riordan groups, which are generated by multiplication and composition operators, to the bi-infinite context. See [5] and [6]. Bi-infinite bloc Toeplitz matrices have been studied for a long time. See [7] and the references therein. In [9] we used doubly infinite matrices to obtain results about groups of generalized Pascal matrices. A multimatrix version of doubly infinite matrices was used in [8] to study Lagrange inversion formulas. A good reference for the theory of formal power series and formal Laurent series is Henrici s boo [2]. In the present paper, we introduce a new way to deal with doubly infinite matrices and show that it is a convenient and effective way to obtain algebraic properties of such matrices. The basic idea is quite simple: we consider a doubly infinite matrix as the sum of its diagonal submatrices. Such idea was motivated by our *Corresponding Author: Luis Verde-Star: Department of Mathematics, Universidad Autónoma Metropolitana, Iztapalapa, Ciudad de México, verde@xanum.uam.mx María Ivonne Arenas-Herrera: Department of Mathematics, Universidad Autónoma Metropolitana, Iztapalapa, Ciudad de México Open Access. 217 María Ivonne Arenas-Herrera and Luis Verde-Star, published by De Gruyter Open. This wor is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4. License.

2 Laurent series and infinite matrices 251 wor in the matrix approach to orthogonal polynomials introduced in [1], where we considered equations in infinite matrices that were solved by finding the diagonals of the unnown matrices in a recursive way. The results obtained for doubly infinite matrices yield immediately some related results for infinite matrices with indices in the nonnegative integers, by means of bloc decomposition and truncation. For example, some results in [11] can be obtained in this way. In Section 2 we present the basic definitions and notation that we will use in this paper. In Section 3 we introduce the representation by diagonals of the Hessenberg doubly infinite matrices, which allows us to consider the ring of Hessenberg doubly infinite matrices as a ring of Laurent series in one indeterminate S with coefficients that do not commute with S. We also show how to use the diagonal representation to compute matrix multiplications and to find inverses of some matrices. In Section 4 we use the diagonal representation in order to obtain some properties related with similarity and commutativity of the doubly infinite matrices. 2 Doubly infinite matrices In this section we define a convergence-free ring of doubly infinite matrices and present the definitions and notation that we will use in this article. Let R be an associative ring with unit, denoted by e. Let L be the set of all bilaterally infinite matrices A = [a n, ], where the indices n and run over all the integers and, either A = or there exists an integer i(a), called the index of A, such that i(a) = min {n : (n, ) Z Z, a n, = }. This means that the support of the map (n, ) a n,, from Z Z to R, is contained in the half-space {(n, ) Z Z : n i(a)}. We say that the entries a n, are on the diagonal of index j if n = j. A matrix is diagonal if all of its nonzero entries, if there are any, lie on a single diagonal. Let D m be the set of diagonal matrices of index m, for m in Z. The zero matrix is in D m for every m. The matrices in L are represented in the usual way. The first subindex corresponds to rows and increases as we go downward, and the second subindex corresponds to columns and increases as we go to the right. Therefore, if A is a nonzero element of L then all the entries a n, above the diagonal of index i(a) are zero and there is at least one nonzero entry in the diagonal of index i(a). We say that L is the set of lower semi-matrices. They are also called infinite Hessenberg matrices. It is easy to verify that L is a ring with the usual addition and multiplication operations. If A = [a n, ] and B = [b n, ] are elements of L and AB = C = [c n, ] then we have c n, = j=+i(b) a n,j b j,, (2.1) and this shows that c n, may be nonzero only when n i(a) + i(b), that is, i(c) i(a) + i(b). Let us note that the multiplication of two elements of L involves only finite sums of elements of R and therefore no convergence problems arise. Köthe and Toeplitz [4] considered some convergence free rings that are larger that L. The identity for the multiplication is denoted by I. It is the element of D such that its (, ) entry is equal to e for every in Z. We show next that the elements of L are connected with formal Laurent series. Let V be the set of infinite column vectors v = (v j ) such that v j R for each j Z and, either v = or there exists an integer i(v) such that v j = if j < i(v) and v i(v) =. Note that every element A of L has its columns in V and that the map v Av is a well-defined additive map on the set V, where (Av) n = =i(v) a n, v. (2.2) Each element v = (v j ) of V can be identified with the formal Laurent series v(z) = v j z j, (2.3) j=i(v)

3 252 María Ivonne Arenas-Herrera and Luis Verde-Star where z is an indeterminate and the coefficients are in R. Therefore Av can be identified with the Laurent series Av(z) = a n, v z n, (2.4) and for each in Z the series n i(a)+i(v) =i(v) A ez = n i(a)+ a n, z n (2.5) corresponds to the -th column of A, and we can consider A as a doubly infinite sequence of formal Laurent series with coefficients in R. It is clear that the rows of the elements of L can be considered as reversed formal Laurent series. See [2]. The shift matrix S is the element of D 1 such that its entry ( + 1, ) is equal to e for every in Z. It is easy to verify that S is invertible and its inverse S 1 is the element of D 1 with entry (, + 1) equal to e for every in Z. If A is an element of L then SA is obtained by translating A one position downwards, AS is obtained translating A one position to the left, S 1 A is equal to A translated one position upwards, and AS 1 is equal to A translated one position to the right. Therefore, for m in Z the product S m AS m is a translation by m positions of A in the diagonal direction, that is, the entry (n, ) of S m AS m is equal to a n m, m. We will use the notation A [m] = S m AS m and call A [m] the diagonal shift of A by m steps. Note that (AB) [m] = A [m] B [m]. In the following proposition we list some basic properties of elements of the ring L. The proofs are simple computations. Proposition If A L commutes with S (or with S 1 ) then A is constant along its diagonals, that is, A is a Toeplitz matrix. 2. If A commutes with S m for some m = then A is a bloc Toeplitz matrix with blocs of size m m. 3. The set D is closed under addition and multiplication and contains the identity I. 4. An element U = [u n, ] of D is invertible if and only if u, is invertible in R for every in Z, and in that case U 1 is also in D. 5. If W is in D m then W = S m U where U = S m W and U is in D. 6. An element W of D m is invertible if and only if its entry ( + m, ) is invertible in R for every in Z, and in that case W 1 is in D m. 7. If A and B are in L and m and are integers then S m AS B = S m+ A [ ] B. 3 Representation by diagonals and non-commutative Laurent series Let A be an element of L and m be an integer. We denote by δ m (A) the element of D m that has a +m, as its entry in the ( + m, ) position for each in Z, that is, the diagonal matrix of index m that coincides with A on the m-th diagonal. Then we can express a nonzero matrix A in the form A = δ (A). (3.1) i(a) This is just a convenient way to say that A is constructed by putting its diagonals in the proper places. Note that there are really no sums involved in (3.1), since for any pair of integers (n, ) there is only one diagonal that may have a nonzero entry in the position (n, ) and it is δ n (A). Now define γ m (A) = S m δ m (A), A L, m Z. (3.2) Note that γ m (A) is in the ring D for every m and δ m (A) = S m γ m (A). Therefore A = S γ (A), A L. (3.3) i(a)

4 Laurent series and infinite matrices 253 We say that this is the representation by diagonals of the matrix A. Therefore the ring of matrices L is the set of matrices of the form A = S U, U D, U i(a) =, (3.4) i(a) together with the zero matrix. This representation for the elements of L loos lie a Laurent series where S plays the role of the indeterminate and the matrices U are the coefficients, but, in general S does not commute with the U. We find next an expression for the multiplication in L using the representation by diagonals. Theorem 3.1. Let A be given by (3.4) and let B = S V, V D, V i(b) =. i(b) Then AB = n i(a)+i(b) S n =i(b) U [ ] n V. (3.5) Proof: By Proposition 2.1, part 7, we have S n U n S V = S n U [ ] n V, n, Z, (3.6) where U [ ] n is the diagonal shift of U n by steps and satisfies (U [ ] n ) j,j = (U n ) j+,j+ for j Z. The limits of the sum over in (3.5) are obtained from i(b) and n i(a). The ring D, which is a subring of L, is isomorphic to the set of functions from Z to the ring R, equipped with its natural operations of addition and multiplication of functions. Note that this ring contains many divisors of zero and that it is commutative only when R is commutative. From Theorem 3.1 we see that L is a noncommutative modification of the ring of formal Laurent series with coefficients that are functions from Z to R. If we consider S as a ind of indeterminate then the fact that, in general, S does not commute with the coefficients, which are in D, is what maes the diagonal shifts appear in the multiplication formula (3.5). The Laurent series representation gives us immediately a sufficient condition for invertibility in L. Proposition 3.1. Let A = i(a) S U, where the matrices U are in D, and suppose that U i(a) is invertible. Then A is invertible in L. First proof: Let B = i(a) S V, V D. We will show that from the equation AB = I we can obtain all the matrices V in a recursive way. By Theorem 3.1 we have AB = S n U [ ] n V. n = i(a) If AB = I then the summand corresponding to n = in the previous sum must be equal to the identity matrix I. That is, U [i(a)] V i(a) i(a) = I. Therefore V i(a) must be equal to the inverse of U [i(a)], which exists since U i(a) i(a) is invertible by the hypothesis. For n > the coefficient of S n in the expression for AB must be zero and therefore n 1 i(a) U [i(a) n] V i(a) = U [ ] n V. = i(a) Since U i(a) is invertible, so are its diagonal shifts, and then the previous equation gives V in terms of V, with i(a) n 1 i(a), and diagonal shifts of some coefficient matrices of A. Therefore, all the diagonal

5 254 María Ivonne Arenas-Herrera and Luis Verde-Star coefficient matrices V are uniquely determined in a recursive way. Using forward substitution it is easy to find an explicit, but quite long, expression for the matrices V. For example, V i(a)+1 = V i(a) U [i(a)] i(a)+1 V i(a), and V i(a)+2 = V i(a) U [i(a)] i(a)+1 V i(a)u [i(a)] i(a)+1 V i(a) V i(a) U [i(a)] i(a)+2 V i(a). If instead of the equation AB = I we consider BA = I, using induction and some straightforward computations we can show that the coefficient matrices of B obtained in this case coincide with the corresponding coefficient matrices of B obtained above. Alternatively, we can show that the matrix B obtained above satisfies BA = I and therefore B is the unique two-sided inverse of A. This completes our first proof. For the usual formal Laurent series the result analogous to this theorem is often proved by using geometric series. Such method of proof also wors in our context, as we show next. Second proof: Let B = S i(a) AUi(A) 1. Then B = n Sn U n+i(a) Ui(A) 1 and thus B = I + C where C = n 1 Sn V n and the coefficient matrices V n = U n+i(a) Ui(A) 1 are in D. Then the positive powers of C have increasing indices and consequently they are summable. Therefore the geometric series F = n ( C)n is well defined and it is clearly a two-sided inverse of B and therefore it is the unique two-sided inverse of B. Now, since A = S i(a) BU i(a), it is easy to see that Ui(A) 1 FS i(a) is the unique two-sided inverse of A. The condition for invertibility of the previous proposition is not necessary. Consider the following simple example. Let R be the ring of 2 2 matrices over R and let [ ] [ ] 1 a =, b =. 1 The elements a and b are divisors of zero, since a 2 = b 2 =, and ab + ba = e, the identity in R. Let A = Ia + Sb. Note that A is a Toeplitz matrix. A simple computation shows that Ib + S 1 a is the inverse of A. Bloc decomposition of the infinite matrices explains why this example wors and also why it is not easy to find necessary conditions for invertibility in L. 4 Some properties of the ring L In this section we use the diagonal representation of the elements of L in order to obtain some properties related with similarity and commutativity. Theorem 4.1. Let A = S U, U D, i(a) and suppose that i(a) = and every entry of U i(a) is an invertible element of the center of R. Then there exists an invertible matrix P of index zero such that P 1 AP = S i(a) U i(a). Proof: Note that U i(a) is an invertible element of the center of the ring D. Let P = S V, where the coefficient matrices V are in D. Consider the equation AP = PS i(a) U i(a). We will find matrices V that satisfy this equation and such that P is invertible. By the multiplication formula (3.5) the equation AP = PS i(a) U i(a) becomes n i(a) S n = U [ ] n V = S r+i(a) V [ i(a)] U i(a). r Equating corresponding coefficients of S r+i(a) in both sides of the previous equation we obtain r U [ ] r +i(a) V = V r [ i(a)] U i(a). (4.1) =

6 Laurent series and infinite matrices 255 When r = we get U i(a) V = V [ i(a)] U i(a), which reduces to V = V [ i(a)], since U i(a) is an invertible element of the center of D. Therefore every arbitrary choice of i(a) invertible elements of R yields a solution for V. One obvious solution is V = I. If r 1 then equation (4.1) can be written as i(a) V r 1 r + U [ ] r +i(a) V = V r [ i(a)] U i(a). U [ r] = If V, V 1,..., V r 1 have been determined then the previous equation can be solved for V r and we get an expression that depends on the diagonal shift V r [ i(a)], the already nown V, and some of the coefficient matrices U of A. Note that P is not unique, since we can choose i(a) arbitrary entries of V in each step. If the matrix U i(a) of the previous theorem is equal to the identity matrix I then A = PS i(a) P 1 and therefore the matrix B = PSP 1 satisfies B i(a) = A. That is, B is a root of order i(a) of A. Note that B depends on P and P is not unique. For matrices of index zero we have the following result. Theorem 4.2. Let A = S U, U D, and suppose that U is an element of the center of D and U U [m] is invertible for every nonzero integer m. Then there exists an invertible matrix P of index zero such that P 1 AP = U. Proof: Let P = S V with V = I. The equation AP = PU is equivalent to n S n n = Equating the coefficients of S n in both sides we obtain U [ n] = U [ ] n V = n V n U. n 1 V n + U [ ] n V = V n U = U V n. This equation holds trivially if n = and, by the hypothesis, for n 1 it can be written as n 1 V n = (U U [ n] ) 1 U [ ] n V, and therefore the coefficient matrices V can be computed in a recursive way. Let us note that the diagonal matrix U of the previous theorem is not necessarily invertible. It may have one entry equal to zero in the diagonal of index zero. Consider, for example, (U ), = for Z, with R = R. Theorem 4.3. Let U be an element of the center of the ring D such that for every nonzero integer m the matrix U U [m] has no zero divisors in its diagonal of index zero. Then if B is an element of L that commutes with U we must have B D. Proof: Let B = i(b) S V, where the V are in D. If UB = BU then = S U [ ] V = S V U. i(b) Therefore U [ ] V = V U = UV for i(b). If = we get (U U [ ] )V = and then V =, since U U [ ] has no zero divisors in the diagonal of index zero. Therefore B = V and it is in D. The following theorem generalizes Proposition 2.1, part2. i(b)

7 256 María Ivonne Arenas-Herrera and Luis Verde-Star Theorem 4.4. Let m be a nonzero integer and U an invertible element of D. Let B = i(b) S V, where the V are in D. If B commutes with S m U then V [ m] = U [ ] V U 1 for i(b). Proof: If BS m U = S m UB then and then S V S m U = S m US V, i(b) and this completes the proof. The equation V [ m] = U [ ] V U 1 means that i(b) S +m V [ m] U = S m+ U [ ] V, i(b), (V ) j+m,j+m = (U) j+,j+ (V ) j,j (U 1 ) j,j, j Z. When U is the identity matrix this equation reduces to (V ) j+m,j+m = (V ) j,j and then the diagonals of B are periodic and B is a bloc Toeplitz matrix. Therefore, in the general case each diagonal of B is determined by any m consecutive entries by the previous equation and we can consider B as some ind of generalized bloc Toeplitz matrix. We show next that the elements of L can also be expressed as Laurent series with a weighted shift instead of the shift S. Let W be an invertible element of D and define the weighted shift T with weight W by T = SW. Then T 2 = SWSW = S 2 W [ 1] W and T 3 = S 3 W [ 2] W [ 1] W. We can find an expression for T as follows. Define W () = I and for 1 define W () = W [ +1] W [ +2] W [ 2] W [ 1] W. This is called the ascending factorial power of W. Then we have T = S W () for. Therefore, for 1 we have T = W 1 () S = S S W 1 () S = S (W 1 () )[]. Define W ( ) = (W 1 () )[] for 1. With this notation we have T = S W () for Z. Theorem 4.5. Let A = i(a) S U, with the U in D. Let W be an invertible element of D and T = SW. Then A = i(a) T V, where V = W 1 () U for i(a). Proof: T V = T W 1 () U = S U for i(a). We consider next some useful maps on the ring L related with commutators. For every A in L we define the Pincherle derivative of A by A = AS SA. It is easy to show that the Pincherle derivative is a derivation on L, that is, (AB) = AB + A B, A, B L. Note that the Pincherle derivative of any Toeplitz matrix is equal to zero. If A = i(a) S U then a simple computation gives us A = i(a) S +1 (U [ 1] U ). Let us note that the (j, j) entry of U [ 1] U is equal to (U ) j+1,j+1 (U ) j,j for j Z. Define the differentiation matrix D by D = S 1 N, where N D and (N) j,j = (j + 1)e. Note that D is not invertible, since (N) 1, 1 =. The Pincherle derivative of D is D = DS SD = S 1 NS N = N [ 1] N = I. Using induction we can see that (D m ) = md m 1 for m 1. The Pincherle derivative of the second ind is defined by A = DA AD, A L.

8 Laurent series and infinite matrices 257 It is a derivation on L and it is clear that S = I and (S ) = S 1 for in Z. A simple computation gives us and then, for A = i(a) S U, we have (S U) = S 1 (NU U [1] N + U), U D, A = i(a) S 1 (N(U U [1] ) + U ). Notice that both inds of derivatives of A L involve differences U U [1] of the coefficient matrices of A. 5 Final remars We have shown that the diagonal representation is a useful tool to deal with the elements of L, for computations and also to derive general properties. Here we presented a few examples, but it is clear that many other algebraic properties of the infinite matrices can be obtained with our methods. Note that the ring R can be, for example, the algebra of n n matrices over some field. In the ring L, using suitable choices of R, we can study orthogonal Laurent polynomials, generalized Riordan groups, Lagrange inversion formulas, and Umbral Calculus. We have not considered here the duality induced by the reflection of the matrices with respect to the anti-diagonal n + = 1. The importance of that duality has been proved in [1], [8], and [9]. The extension to the case of multi-matrices (indices in Z d ) is a subject for further research. See [8], where Lagrange inversion in several variables was studied in a ring of multi-matrices. Acnowledgement: Research partially supported by grant 2263 from CONACYT, México. References [1] M. Barnabei, A. Brini, G. Nicoletti, Recursive matrices and Umbral Calculus, J. of Algebra, 75 (1982) [2] P. Henrici, Applied and computational Complex Analysis, Vol. I, J. Wiley, N.Y [3] E. Jabotinsy, Analytic iteration, Trans. Amer. Math. Soc. 18 (1963) [4] G. Köthe, O. Toeplitz, Theorie der halbfiniten unendlichen Matrizen, J. Reine Angew. Math. 165 (1931) [5] A. Luzón, D. Merlini, M.A. Morón, R. Sprugnoli, Identities induced by Riordan arrays, Linear Algebra Appl. 436 (212) [6] A. Luzón, M.A. Morón, J.L. Ramírez, Double parameter recurrences for polynomials in bi-infinite Riordan matrices and some derived identities, Linear Algebra Appl. 511 (216) [7] C.V.M. van der Mee, S. Seatzu, G. Rodriguez, Spectral factorization of bi-infinite multi-index bloc Toeplitz matrices, Linear Algebra Appl. 343/344 (22) [8] L. Verde-Star, Dual operators and Lagrange inversion in several variables, Adv. in Math. 58 (1985) [9] L. Verde-Star, Groups of generalized Pascal matrices, Linear Algebra Appl. 382 (24) [1] L. Verde-Star, Characterization and construction of classical orthogonal polynomials using a matrix approach, Linear Algebra Appl. 438 (213) [11] L. Verde-Star, Polynomial sequences generated by infinite Hessenberg matrices, Spec. Matrices 217; 5:64 72.