Compatibility and companions for Leonard pairs

Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD–D canonical form and the LB–UB canonical form

We first summarize the basic structure of the outer distribution module of a completely regular code. Then, employing a simple lemma concerning eigenvectors in association schemes, we propose to study the tightest case, where the indices of the eigenspace that appear in the outer distribution module are equally spaced. In addition to the arithmetic codes of the companion paper, this highly structured class includes other beautiful examples and we propose the classification of $Q$-polynomial completely regular codes in the Hamming graphs. A key result is Theorem 3.10 which finds that the $Q$-polynomial condition is equivalent to the presence of a certain Leonard pair. This connection has impact in two directions. First, the Leonard pairs are classified and we gain quite a bit of information about the algebraic structure of any code in our class. But also this gives a new setting for the study of Leonard pairs, one closely related to the classical one where a Leonard pair arises from each thin/dual-thin irreducible module of a Terwilliger algebra of some $P$- and $Q$-polynomial association scheme, yet not previously studied. It is particularly interesting that the Leonard pair associated to some code $C$ may belong to one family in the Askey scheme while the distance-regular graph in which the code is found may belong to another.

In Linear algebra, the concept of Leonard pair (LP) was motivated by the theory of Q-polynomial distance-regular graphs. In this dissertation, we will first give a brief introduction to LPs and to two closely-related classes of objects: (i) bipartite Leonard pairs (BLPs) and (ii) almost bipartite Leonard pairs (ABLPs). Taking these as departure points, we will introduce a new class of object - doubly almost bipartite Leonard pairs (DABLPs). The primary aim of our work is to fully classify (up to isomorphism) this new family. In addition, since there is known to be a natural correspondence between Leonard pairs and families of orthogonal polynomials, we reveal which families of orthogonal polynomials correspond to the DABLPs. Several related objects, such as Leonard triples, modular Leonard triples, spin Leonard pairs, and near-bipartite Leonard pairs have corresponding notions for the doubly almost bipartite case. These analogous objects are also defined and briefly explored.

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations and that satisfy (i) and (ii) below. There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing is diagonal. There exists a basis for V with respect to which the matrix representing is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. In the literature, there are some parameters that are used to describe Leonard pairs called the intersection numbers , , , and the dual eigenvalues . In this paper, we provide two characterizations of Leonard pairs. For the first characterization, the focus is on the and . For the second characterization, the focus is on the , , and .

The universal DAHA of type [formula omitted] and Leonard pairs of q-Racah type

A characterization of Leonard pairs using the parameters [formula omitted

It is a standard and very useful technique to approximate geometric objects and maps by linear objects and maps. A principal (trivial) example in basic analysis is approximating an open subspace U of \(\mathbb{K}^{n}\) at a point by the linear space \(\mathbb{K}^{n}\) itself by visualizing \(\mathbb{K}^{n}\) as a tangent space to U at the given point. Another (less trivial) example is the derivative of a differentiable map f on U in a point that is the best approximation of f nearby this point by a linear map. In the first section we extend this technique to premanifolds and morphisms of premanifolds. Thus we first define the tangent space of a premanifold at a point, which will be a \(\mathbb{K}\)-vector space that we visualize as the ”best linear approximation“ of the premanifold at the given point. Then we define the derivative in a point x of a morphism f. This will be a linear map from tangent space at x to the tangent space at \(f(x)\).

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

The almost split triangles for perfect complexes over gentle algebras

This goal of this chapter is to find finitely many canonical representatives of each similarity class of square matrices with entries in a field and correspondingly of each isomorphism class of linear maps from a finite-dimensional vector space to itself.Section 1 frames the problem in more detail. Section 2 develops the theory of determinants over a commutative ring with identity in order to be able to work easily with characteristic polynomials det(?I — A). The discussion is built around the principle of "permanence of identities," which allows for passage from certain identities with integer coefficients to identities with coefficients in the ring in question.Section 3 introduces the minimal polynomial of a square matrix or linear map. The Cayley-Hamilton Theorem establishes that such a matrix satisfies its characteristic equation, and it follows that the minimal polynomial divides the characteristic polynomial. It is proved that a matrix is similar to a diagonal matrix if and only if its minimal polynomial is the product of distinct factors of degree 1. In combination with the fact that two diagonal matrices are similar if and only if their diagonal entries are permutations of one another, this result solves the canonical-form problem for matrices whose minimal polynomial is the product of distinct factors of degree 1.Section 4 introduces general projection operators from a vector space to itself and relates them to vector-space direct-sum decompositions with finitely many summands. The summands of a directsum decomposition are invariant under a linear map if and only if the linear map commutes with each of the projections associated to the direct-sum decomposition.Section 5 concerns the Primary Decomposition Theorem, whose subject is the operation of a linear map L: V ? V with V finite-dimensional. The statement is that if L has minimal polynomial \( P_1 (\lambda )^{l_1 } \cdots P_k (\lambda )^{l_k } \) with the Pj (?) distinct monic prime, then V has a unique direct-sum decomposition in which the respective summands are the kernels of the linear maps \( P_j (L)^{l_j } \), and moreover the minimal polynomial of the restriction of L to the j th summand is \( P_j (\lambda )^{l_j } \).Sections 6–7 concern Jordan canonical form. For the case that the prime factors of the minimal polynomial of a square matrix all have degree 1, the main theorem gives a canonical form under similarity, saying that a given matrix is similar to one in "Jordan form" and that the Jordan form is completely determined up to permutation of the constituent blocks. The theorem applies to all square matrices if the field is algebraically closed, as is the case for C. The theorem is stated and proved in Section 6, and Section 7 shows how to make computations in two different ways.

In this closing chapter we will be engaged again with notions and techniques typical for the field of Linear Algebra, but here our attention will also be focused on the abstract structure of vector spaces and linear mappings (or linear transformations) between them. Vectors—the elements of a vector space—do not need to be ordered lists of numbers, polynomials can also be viewed as vectors. Of course, matrices continue to play a prominent role, namely as representations of linear mappings between vector spaces with predetermined bases.KeywordsVector SpaceSingular Value DecompositionMatrix RepresentationCharacteristic PolynomialNull SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let Vbe a vector space of matrices over a field and ka fixed positive integer. In this chapter we first survey results concerning linear maps on certain types of Vthat preserve one of the following:(a) the set of rank kmatrices, (b) the set of matrices of rank less than k. We next survey results concerning linear maps on certain symmetry classes of tensors that preserve nonzero decomposable elements.

On the Hyers-Ulam Stability of Linear Mappings

In this note, we give a simple proof as well as an extension of a very recent result of B. Zheng, J. Xu and A. Fosner concerning linear maps between vector spaces of complex square matrices that preserve the rank of tensor products of matrices by using a structure theorem of R. Westwick on linear maps between tensor product spaces that preserve non-zero decomposable elements.

Linear maps on symmetric spaces: the invariance of nonzero decomposable elements