A multilinear algebra proof of the Cauchy–Binet formula and a multilinear version of Parsevalʼs identity

We prove the Cauchy–Binet determinantal formula using multilinear algebra by first generalizing it to an identity not involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of the classical Parseval identity.     

A general product of tensors with applications

We study a general product of two n  -dimensional tensors A$\mathbb{A}$ and B$\mathbb{B}$ with orders m?2$m?2$ and k?1$k?1$. This product satisfies the associative law, and is a generalization of the usual matrix product. Using this product, many concepts and known results of tensors can be simply expressed and/or proved, and a number of applications of it will be given. Using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables, we define the similarity and congruence of tensors (which are also the generalizations of the corresponding relations for matrices), and prove that similar tensors have the same characteristic polynomials, thus the same spectra. We study two special kinds of similarity: permutational similarity and diagonal similarity, and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors. We also define the direct product of tensors (in matrix case it is also called the Kronecker product), and give its applications in the study of the spectra of two kinds of the products of hypergraphs. We also give applications of this general product in the study of nonnegative tensors, including a characterization of primitive tensors, the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors.     

Existence of real eigenvalues of real tensors

We use the Brouwer degree to establish the existence of real eigenpairs of higher order real tensors in various settings. Also, we provide some finer criteria for the existence of real eigenpairs of two-dimensional real tensors and give a complete classification of the Brouwer degree zero and ±2 maps induced by general third order two-dimensional real tensors.     

Quantum Pfaffians and hyper-Pfaffians

The concept of the quantum Pfaffian is rigorously examined and refurbished using the new method of quantum exterior algebras. We derive a complete family of Plücker relations for the quantum linear transformations, and then use them to give an optimal set of relations required for the quantum Pfaffian. We then give the formula between the quantum determinant and the quantum Pfaffian and prove that any quantum determinant can be expressed as a quantum Pfaffian. Finally the quantum hyper-Pfaffian is introduced, and we prove a similar result of expressing quantum determinants in terms of quantum hyper-Pfaffians at modular cases.     

The work of Gian-Carlo rota on invariant theory

The purpose of this paper is twofold: first, to explain Gian-Carlo Rotas work on invariant theory; second, to place this work in a broad historical and mathematical context. Rotas work falls under three specific cases: vector invariants, the invariants of binary forms, and the invariants of skew-symmetric tensors. We discuss each of these cases and show how determinants and straightening play central roles. In fact, determinants constitute all invariants in the vector case; for binary forms and skew-symmetric tensors, they constitute all invariants when invariants are represented symbolically. Consequently, we explain the symbolic method both for binary forms and for skew-symmetric tensors, where Rota developed generalizations of the usual notion of a determinant. We also discuss the Grassmann algebra, with its two operations of meet and join, which was a theme which ran through Rotas work on invariant theory almost from the very beginning.To the memory of Gian-Carlo Rota     

An analogue of the Harer-Zagier formula for unicellular maps on general surfaces

A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is simply connected. In a famous article, Harer and Zagier established a formula for the generating function of unicellular maps counted according to the number of vertices and edges. The keystone of their approach is a counting formula for unicellular maps on orientable surfaces with n edges, and with vertices colored using every color in [q] (adjacent vertices are authorized to have the same color). We give an analogue of this formula for general (locally orientable) surfaces.Our approach is bijective and is inspired by Lass?s proof of the Harer-Zagier formula. We first revisit Lass?s proof and twist it into a bijection between unicellular maps on orientable surfaces with vertices colored using every color in [q], and maps with vertex set [q] on orientable surfaces with a marked spanning tree. The bijection immediately implies Harer-Zagier?s formula and a formula by Jackson concerning bipartite unicellular maps. It also shed a new light on constructions by Goulden and Nica, Schaeffer and Vassilieva, and Morales and Vassilieva. We then extend the bijection to general surfaces and obtain a correspondence between unicellular maps on general surfaces with vertices colored using every color in [q], and maps on orientable surfaces with vertex set [q]with a marked planar submap. This correspondence gives an analogue of the Harer-Zagier formula for general surfaces. We also show that this formula implies a recursion formula due to Ledoux for the numbers of unicellular maps with given numbers of vertices and edges.     

Fredholm Properties of the Difference of Orthogonal Projections in a Hilbert Space

Buckholtz (Proc. Amer. Math. Soc. 128 (2000), 1415–1418) gave necessary and sufficient conditions for the invertibility of the difference of two orthogonal projections in a Hilbert space. We generalize this result by investigating when the difference of such projections is a Fredholm operator, and give an explicit formula for its Fredholm inverse.     

A fast algorithm for the spectral radii of weakly reducible nonnegative tensors

In this paper, we propose a fast algorithm for computing the spectral radii of symmetric nonnegative tensors. In particular, by this proposed algorithm, we are able to obtain the spectral radii of weakly reducible symmetric nonnegative tensors without requiring the partition of the tensors. As we know, it is very costly to determine the partition for large‐sized weakly reducible tensors. Numerical results are reported to show that the proposed algorithm is efficient and also able to compute the spectral radii of large‐sized tensors. As an application, we present an algorithm for testing the positive definiteness of Z‐tensors. By this algorithm, it is guaranteed to determine the positive definiteness for any Z‐tensor.     

Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups

Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a q-analog of Euler?s exponential generating function formula for the Eulerian numbers (Shareshian and Wachs, 2010 [17]). They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the wreath product of the cyclic group with the symmetric group, also known as the group of colored permutations. We use this group to introduce colored Eulerian quasisymmetric functions, which are a generalization of Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, which reduces to a formula of Shareshian and Wachs for the Eulerian quasisymmetric functions. We show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics, which generalize the Shareshian–Wachs q-analog of Euler?s formula, formulas of Foata and Han, and a formula of Chow and Gessel.     

A concise proof of Kruskal’s theorem on tensor decomposition

A theorem of J. Kruskal from 1977, motivated by a latent-class statistical model, established that under certain explicit conditions the expression of a third-order tensor as the sum of rank-1 tensors is essentially unique. We give a new proof of this fundamental result, which is substantially shorter than both the original one and recent versions along the original lines.     

Further Results on the Drazin Inverse of Tensors via Einstein Product

In this paper,we give further results on the Drazin inverse of tensors via the Einstein product.We give a limit formula for the Drazin inverse of tensors.By using this formula,the representations for the Drazin inverse of several block tensor are obtained.Further,we give the Drazin inverse of the sum of two tensors based on the representation for the Drazin inverse of a block tensor.     

ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ? Λ^d(F ^k) and s ? Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.     

Two-dimensional representations of finitely generated free group over an arbitrary field

Working over an arbitrary field, we give equivalent conditions for a representation of a finitely generated free group with given traces and determinants to exist and to be reducible, respectively; also, we classify all two-dimensional representations of a finitely generated free group.     

Factorization strategies for third-order tensors

Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-1 outer products using either the CANDECOMP/PARAFAC (CP) or the Tucker models, or some variation thereof. Such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array. The specifics of the approximate representation (such as how many terms to use in the sum, orthogonality constraints, etc.) depend on the application.In this paper, we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem. Reminiscent of matrix factorizations, we present a new factorization of a tensor as a product of tensors. To derive the new factorization, we define a closed multiplication operation between tensors. A major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications.Specifically, this new multiplication allows us to introduce concepts such as tensor transpose, inverse, and identity, which lead to the notion of an orthogonal tensor. The multiplication also gives rise to a linear operator, and the null space of the resulting operator is identified. We extend the concept of outer products of vectors to outer products of matrices. All derivations are presented for third-order tensors. However, they can be easily extended to the order-p(p>3) case. We conclude with an application in image deblurring.     

On reducible monodromy representations of some generalized Lamé equation

In this note, we compute the explicit formula of the monodromy data for a generalized Lamé equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemann–Hilbert problem.

     

Operator extensions of Hua’s inequality

We give an extension of Hua’s inequality in pre-Hilbert C^∗-modules without using convexity or the classical Hua’s inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C^∗-modules, another extension of Hua’s inequality is obtained. We also present an operator Hua’s inequality, which is equivalent to operator convexity of given continuous real function.     

Representation of tripotents and representations via tripotents

Let A be an algebra. An element A∈A is called tripotent if A³=A. We study the questions: if both A and B are tripotents, then: Under what conditions are A+B and AB tripotent? Under what conditions do A and B commute? We extend the partial order from the Hilbert space idempotents to the set of all tripotents and show that every normal tripotent is self-adjoint. For A=M_n(C) we describe the set of all finite sums of tripotents, the convex hull of tripotents and the set of all tripotents averages. We also give the new proof of rational trace matrix representations by Choi and Wu [2].     

Subtracting a best rank-1 approximation may increase tensor rank

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R?2 and k?3. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for real-valued 2×2×2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2×2×2 tensors (which have rank 2 or 3), subtracting a best rank-1 approximation results in a tensor that has rank 3 and lies on the boundary between the rank-2 and rank-3 sets. Hence, for a typical tensor of rank 2, subtracting a best rank-1 approximation increases the tensor rank.     

Fischer determinantal inequalities and Highamʼs Conjecture

We obtain a Fischer type determinantal inequality for matrices with given angular numerical range. We discuss the growth factor for Gaussian elimination for linear systems in which the coefficient matrix has this form and give a proof of Higham?s Conjecture.