Pair Of Linear Transformations Research Articles

The horizontal tensor complementarity problem

This article explores a new type of nonlinear complementarity problem, namely the horizontal tensor complementarity problem (HTCP), which is a natural extension of the horizontal linear complementarity problem studied in Sznajder [Degree-theoretic analysis of the vertical and horizontal linear complementarity problems [Ph.D. Thesis]. University of Maryland Baltimore County; 1994]. We extend the concepts of R 0 , R and P pairs from a pair of linear transformations given in Chi et al. [The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra. J Global Optim. 2019;73:153–169] to a pair of tensors. When a given pair of tensors has these properties, we use degree-theoretic tools to discuss the existence and boundedness of solutions to the HTCP. Finally, we study a uniqueness result of the solution of the HTCP.

Classical Leonard pairs having LB-TD form

ABSTRACTLet denote an algebraically closed field, and let V denote a vector space over with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations on V such that, for each of , there exists a basis for V with respect to which the matrix representing that linear transformation is diagonal and the matrix representing the other linear transformation is irreducible tridiagonal. Assume that the dimension of V is at least 4. In this paper, we consider a class of Leonard pairs said to be of classical type that consists of four families: Racah type, Hahn type, dual-Hahn type and Krawtchouk type. Let be a classical Leonard pair on V . By using Askey-Wilson relations of Leonard pairs and representation theory of universal enveloping algebra , we show that is of either Racah type or Krawtchouk type if and only if there exists a basis for V with respect to which the matrices representing are respectively lower bidiagonal with subdiagonal entries all 1 and irreducible tridiagonal.

Totally almost bipartite Leonard pairs and Leonard triples of q-Racah type

Let denote an algebraically closed field of characteristic zero. Let V denote a vector space over with finite positive dimension. A Leonard pair on V is an ordered pair of linear transformations in such that for each of these transformations there exists a basis for V with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal. Whenever these two tridiagonal matrices are almost bipartite, the Leonard pair is said to be totally almost bipartite. The notion of a Leonard triple and the corresponding notion of totally almost bipartite are similarly defined. Let q denote a quantum parameter of a Leonard pair and let ‘TAB’ be an abbreviation for ‘totally almost bipartite’. In this paper we show that a TAB Leonard pair with q equal to is of Bannai/Ito type, and a TAB Leonard pair with q being not a root of unity is of q-Racah type. Under the assumption that q is not a root of unity, we classify, up to isomorphism, the TAB Leonard pairs of q-Racah type and the TAB Leonard triples of q-Racah type.

Leonard triples of Krawtchouk type

Let denote an algebraically closed field of characteristic 0. Let denote a vector space over of finite positive dimension. A Leonard pair on is an ordered pair of linear transformations in such that for each of these transformations there exists a basis for with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal. The diameter of the Leonard pair is defined as . Let be a Leonard pair of diameter . The Leonard pair is of Krawtchouk type whenever its eigenvalue sequence and its dual eigenvalue sequence are both . The notions of a Leonard triple and a Leonard triple of Krawtchouk type are similarly defined. In this paper, let be a Leonard pair of Krawtchouk type. We first determine all linear transformations in such that is a Leonard triple of Krawtchouk type. Then we classify up to isomorphism Leonard triples of Krawtchouk type.

Transition maps between the 24 bases for a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A ∗ : V → V that satisfy (i) and (ii) below: 1. [(i)] There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A ∗ is diagonal. 2. [(ii)] There exists a basis for V with respect to which the matrix representing A ∗ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V . In an earlier paper we described 24 special bases for V . One feature of these bases is that with respect to each of them the matrices that represent A and A ∗ are (i) diagonal and irreducible tridiagonal or (ii) irreducible tridiagonal and diagonal or (iii) lower bidiagonal and upper bidiagonal or (iv) upper bidiagonal and lower bidiagonal. For each ordered pair of bases among the 24 , there exists a unique linear transformation from V to V that sends the first basis to the second basis; we call this the transition map. In this paper we find each transition map explicitly as a polynomial in A , A ∗ .

The switching element for a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A ∗ : V → V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A ∗ is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A ∗ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Let { v i } i = 0 d (resp. { w i } i = 0 d ) denote a basis for V referred to in (i) (resp. (ii)). We show that there exists a unique linear transformation S : V → V that sends v 0 to a scalar multiple of v d , fixes w 0 , and sends w i to a scalar multiple of w i for 1 ⩽ i ⩽ d . We call S the switching element. We describe S from many points of view.

The split decomposition of a tridiagonal pair

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A ∗ : V → V that satisfy (i)–(iv) below: (i) Each of A, A ∗ is diagonalizable. (ii) There exists an ordering V 0, V 1, …, V d of the eigenspaces of A such that A ∗ V i ⊆ V i - 1 + V i + V i + 1 for 0 ⩽ i ⩽ d, where V -1 = 0, V d+1 = 0. (iii) There exists an ordering V 0 ∗ , V 1 ∗ , … , V δ ∗ of the eigenspaces of A ∗ such that AV i ∗ ⊆ V i - 1 ∗ + V i ∗ + V i + 1 ∗ for 0 ⩽ i ⩽ δ, where V - 1 ∗ = 0 , V δ + 1 ∗ = 0 . (iv) There is no subspace W of V such that both AW ⊆ W , A ∗ W ⊆ W , other than W = 0 and W = V. We call such a pair a tridiagonal pair on V. In this note we obtain two results. First, we show that each of A, A ∗ is determined up to affine transformation by the V i and V i ∗ . Secondly, we characterize the case in which the V i and V i ∗ all have dimension one. We prove both results using a certain decomposition of V called the split decomposition.

Global Representation of Periodic Solution for Vibro-Impact System (Explicit Form Using Implicit Impact Time Parameters)

This paper gives exact and global representation of all periodic solutions appearing in 1-DOF vibro-impacting system. The previously derived form of periodic solution by authors can be represented as superposition of periodic solution for linearized system and finite number of continuous periodic functions which is evaluated by equivalent impulsive responses caused by nonlinearity. However, it is not practical, because determination of impact velocities characterising periodic solutions includes infinite product and summation procedures in the form. To avoid this difficulties, relation between two types of representation of periodic solutions, traditional and newly one by authors, is reconsidered. As a result, it is clarified that transcendental equations for periodic connecting conditions in these representations can be mutually transformed by a pair of linear transformations. According to apply this fact, the finite product and summation scheme determining impact velocities of periodic solutions is derived.

Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A ∗ : V → V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A ∗ is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A ∗ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Let X denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X with respect to the basis (ii) is tridiagonal. We show that X is spanned by I , A , A ∗ , AA ∗ , A ∗ A and these elements form a basis for X provided the dimension of V is at least 3.

Matrix units associated with the split basis of a Leonard pair

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A ∗ : V → V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A ∗ is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A ∗ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. It is known that there exists a basis for V with respect to which the matrix representing A is lower bidiagonal and the matrix representing A ∗ is upper bidiagonal. In this paper we give some formulae involving the matrix units associated with this basis.

The determinant of AA∗– A∗A for a Leonard pair A, A∗

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A: V → V and A ∗: V → V that satisfy (i), (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A ∗ is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A ∗ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. In this paper we investigate the commutator AA ∗ − A ∗ A. Our results are as follows. Abbreviate d=dim V − 1 and first assume d is odd. We show AA ∗ − A ∗ A is invertible and display several attractive formulae for the determinant. Next assume d is even. We show that the null space of AA ∗ − A ∗ A has dimension 1. We display a nonzero vector in this null space. We express this vector as a sum of eigenvectors for A and as a sum of eigenvectors for A ∗.

Some trace formulae involving the split sequences of a Leonard pair

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A ∗ : V → V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A ∗ is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A ∗ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Let diag( θ 0, θ 1, … , θ d ) denote the diagonal matrix referred to in (ii) above and let diag ( θ 0 * , θ 1 * , … , θ d * ) denote the diagonal matrix referred to in (i) above. It is known that there exists a basis u 0, u 1, … , u d for V and there exist scalars ϕ 1, ϕ 2, … , ϕ d in K such that Au i = θ i u i + u i+1 (0 ⩽ i ⩽ d − 1), Au d = θ d u d , A * u i = ϕ i u i - 1 + θ i * u i ( 1 ⩽ i ⩽ d ) , A * u 0 = θ 0 * u 0 . The sequence ϕ 1, ϕ 2, … , ϕ d is called the first split sequence of the Leonard pair. It is known that there exists a basis v 0, v 1, … , v d for V and there exist scalars ϕ 1, ϕ 2, … , ϕ d in K such that Av i = θ d− i v i + v i+1 (0 ⩽ i ⩽ d − 1), Av d = θ 0 v d , A * v i = ϕ i v i - 1 + θ i * v i ( 1 ⩽ i ⩽ d ) , A * v 0 = θ 0 * v 0 . The sequence ϕ 1, ϕ 2, … , ϕ d is called the second split sequence of the Leonard pair. We display some attractive formulae for the first and second split sequence that involve the trace function.

Leonard pairs and the q-Racah polynomials

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A: V→ V and A *: V→ V that satisfy the following two conditions: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A * is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A * is irreducible tridiagonal. Wecall such a pair a Leonard pair on V. In the appendix to [Linear Algebra Appl. 330 (2001), p. 149] we outlined a correspondence between Leonard pairs and a class of orthogonal polynomials consisting of the q-Racah polynomials and some related polynomials of the Askey scheme. We also outlined how, for the polynomials in this class, the 3-term recurrence, difference equation, Askey–Wilson duality, and orthogonality can be obtained in a uniform manner from the corresponding Leonard pair. The purpose of this paper is to provide proofs for the assertions which we made in that appendix.

Leonard Pairs from 24 Points of View

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A: V → V and A*: V → V that satisfy both conditions below: (i) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A* is irreducible tridiagonal. (ii) There exists a basis for V with respect to which the matrix representing A* is diagonal and the matrix representing A is irreducible tridiagonal. We call such a pair a Leonard pair on V. Referring to the above Leonard pair, we investigate 24 bases for V on which the action of A and A* take an attractive form. Our bases are described as follows. Let Ω denote the set consisting of four symbols 0, d, 0*, d*. We identify the symmetric group S 4 with the set of all linear orderings of Ω. For each element g of S 4 , we define an (ordered) basis for V, which we denote by [g]. The 24 resulting bases are related as follows. For all elements wxyz in S 4 , the transition matrix from the basis [wxyz] to the basis [xwyz], (respectively [wyxz]), is diagonal, (respectively lower triangular). The basis [wxzy] is the basis [wxyz] in inverted order. The transformations A and A* act on the 24 bases as follows: For all g ∈ S 4 , let A g , (respectively A* g ), denote the matrix representing A, (respectively A*), with respect to [g]. To describe A g and A* g , we refer to 0*, d* as the starred elements of Ω. Writing g = wxyz, if neither of y, z are starred then A g is diagonal and A* g is irreducible tridiagonal. If y is starred but z is not, then A g is lower bidiagonal and A* g is upper bidiagonal. If z is starred but y not, then A g is upper bidiagonal and A* g is lower bidiagonal. If both of y, z are starred, then A g is irreducible tridiagonal and A* g is diagonal. We define a symmetric binary relation on S 4 called adjacency. An element wxyz of S 4 is by definition adjacent to each of xwyz, wyxz, wxzy and no other elements of S 4 . For all ordered pairs of adjacent elements g, h in S 4 , we find the entries of the transition matrix from the basis [g] to the basis [h]. We express these entries in terms of the eigenvalues of A, the eigenvalues of A*, and two sequences of parameters called the first split sequence and the second split sequence. For all g E S 4 , we compute the entries of A g and A* g in terms of the eigenvalues of A, the eigenvalues of A*, the first split sequence and the second split sequence.

Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A:V→V and A *:V→V satisfying both conditions below: 1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A * is irreducible tridiagonal. 2. [(ii)] There exists a basis for V with respect to which the matrix representing A * is diagonal and the matrix representing A is irreducible tridiagonal. We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A * on V, there exists a sequence of scalars β,γ,γ *,ϱ,ϱ * taken from K such that both 0= [A,A 2A *−βAA *A+A *A 2−γ(AA *+A *A)−ϱA *], 0= [A *,A *2A−βA *AA *+AA *2−γ *(A *A+AA *)−ϱ *A], where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.

On dualizing a multivariable Poisson summation formula

It is known [7] that dualizing a form of the Poisson summation formula yields a pair of linear transformations which map a function o of one variable into a function and its cosine transform in a generalized sense. The present work presents conditions on o for which the transform relation holds in the classical sense, and extends this result to a class of generalizations of the Poisson formula in any number of dimensions.

Direct sums and direct products of canonical pencils of matrices

A theorem of Kulikov characterizes the K[ x]-modules which are direct sums of finite-dimensional (as a K-vector space) indecomposable modules, where K[ x] is the polynomial ring over the field K. In this paper an analogous characterization is given for modules over the ring R, arising from pairs of linear transformations between a pair of complex vector spaces, ( V, W). R is a certain subring of the ring of 3 × 3 complex matrices. The equivalence between the category of right R-modules and the category of systems enables one to work entirely in the category of systems. (A pair of complex vector spaces is a system if and only if there is a C -bilinear map from C 2 × V to W). R-modules that are direct sums of finite-dimensional indecomposable subsystems are called pure-projective. The above characterization of pure-projective R-modules is used to prove that an R-module M is projective if and only if Ext( M, R) = 0. Direct products of finite-dimensional indecomposable R-modules are also studied, and a theorem pinpoints those that are pure-projective. An example of an R-module M that is not pure-projective, but with the property that every finite subset of M is contained in a pure-projective direct summand of M, is given. A by-product of this example is a class of matrices that generalizes the Vandermonde matrices.

A Purity Criterion for Pairs of Linear Transformations

In connection with the study of perturbation methods for differential eigenvalue problems, Aronszajn put forth a theory of systems (X, Y; A, B) consisting of a pair of linear transformations A, B:X → Y (see [1]; cf. also [2]). Here X and Y are complex vector spaces, possibly of infinite dimension. The algebraic aspects of this theory, where no restrictions of topological nature are imposed, where developed in [3] and [5]. We hasten to point out that the category of C2-systems (definition in § 1) in which this algebraic investigation takes place is equivalent to the category of all right modules over the ring of matrices of the form

On Algebraic Equivalence Between Pairs of Linear Transformations

On Algebraic Equivalence Between Pairs of Linear Transformations