Commutativity Of Matrices Research Articles

Decomposition of matrices into products of commutators of quadratic matrices

Let F be a field, p ( x ) be a quadratic polynomial in F [ x ] with a non-zero constant term, and n ≥ 2 be a positive integer. We say that T ∈ M n ( F ) is p ( x ) -quadratic if p ( T ) = 0 . In this paper, given A ∈ SL n ( F ) , we show that A can be decomposed into a product of at most two commutators of p ( x ) -quadratic matrices if either p ( x ) has two distinct roots in F and A is non-scalar, or if F is algebraically closed and has characteristic different from 2. A similar result for matrices over real quaternion division rings is also provided.

A note on products of commutators of quadratic matrices

A note on products of commutators of quadratic matrices

Products of commutators of unipotent matrices of index $2$ in $\mathrm{GL}_n(\mathbb H)$

The aim of this paper is to show that if $\mathbb{H}$ is the real quaternion division ring and $n$ is an integer greater than $1,$ then every matrix in the special linear group $\mathrm{SL}_n(\mathbb{H})$ can be expressed as a product of at most three commutators of unipotent matrices of index $2$.

Integrability in $$[d+1]$$ dimensions: combined local equations and commutativity of the transfer matrices

Integrability in $$[d+1]$$ dimensions: combined local equations and commutativity of the transfer matrices

Generalized matrix functions, permutation matrices and symmetric matrices

The purpose of this paper is to study generalized matrix functions only using the permutation matrices and symmetric matrices. Firstly the zeroness of a generalized matrix function and then the equality of two generalized matrix functions on the permutation matrices and symmetric matrices will be examined. Secondly generalized matrix functions preserving commutativity of the permutation matrices or commutativity of the symmetric matrices will be characterized. Thirdly generalized matrix functions which preserve product of the permutation matrices or product of the symmetric matrices will be investigated. Finally the Cayley-Hamilton Theorem for generalized characteristic polynomials using the permutation matrices and symmetric matrices will be studied.

An Ising-type formulation of the six-vertex model

We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the coordinate Bethe ansatz for the eigenvectors for all higher spin generalizations of the six-vertex model. The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. As a mathematical identity the emerging star-triangle relation is equivalent to the Pfaff-Saalschütz-Jackson summation formula, originally discovered by J. F. Pfaff in 1797. Plausibly, all vertex models associated with quantized affine Lie algebras and superalgebras can be reformulated as Ising-type models with only two-spin interactions.

Commutators of random matrices from the unitary and orthogonal groups

We investigate the statistical properties of C = uvu−1v−1, when u and v are independent random matrices, uniformly distributed with respect to the Haar measure of the groups U(N) and O(N). An exact formula is derived for the average value of power sum symmetric functions of C, and also for products of the matrix elements of C, similar to Weingarten functions. The density of eigenvalues of C is shown to become constant in the large-N limit, and the first N−1 correction is found.

Decomposition of matrices into commutators of unipotent matrices of index 2

Let $\mathbb{C}$ be the complex field. Denote by $\mathrm{SL}_n(\mathbb{C})$ the group of all complex $n\times n$ matrices with determinant $1$. It is proved that every matrix in $\mathrm{SL}_n(\mathbb{C})$ can be decomposed into a product of two commutators of unipotent matrices of index $2$. Moreover, two is the smallest such number.

Stability of Potential Systems to General Positional Perturbations

This paper shows that a potential system with one multiple eigenvalue of multiplicity two or more can assuredly be made unstable by infinitesimal positional perturbations. The explicit nature of these pertubatory forces is provided. It is shown that the matrices that describe these perturbatory forces are not required to commute with the potential matrix. The general positional perturbatory forces that bring about instability in the perturbed potential system are shown to include, as special cases, circulatory forces that do and that do not commute with the potential matrix. The condition on commutativity of matrices is quite stringent, and its removal has a significant effect on generalizing the conditions that lead to instability. The paper expands the generalized Merkin instability result to include more general, noncirculatory positional perturbations, and it eliminates restrictions imposed on the perturbations by commutation requirements. The structure of infinitesimal as well as finite perturbatory matrices that guarantee instability is obtained. Practical implications of the mathematical results to natural and engineered systems are given. It is pointed out that potential systems with two “nearly” equal frequencies of vibration are, in general, susceptible to instability, created by “small” perturbatory forces, where the terms nearly and small are quantified.

Joint numerical ranges and commutativity of matrices

The connection between the commutativity of a family of n×n matrices and the generalized joint numerical ranges is studied. For instance, it is shown that F is a family of mutually commuting normal matrices if and only if the joint numerical range Wk(A1,…,Am) is a polyhedral set for some k satisfying |n/2−k|≤1, where {A1,…,Am} is a basis for the linear span of the family; equivalently, Wk(X,Y) is polyhedral for any two X,Y∈F. More generally, characterization is given for the c-numerical range Wc(A1,…,Am) to be polyhedral for any n×n matrices A1,…,Am. Other results connecting the geometrical properties of the joint numerical ranges and the algebraic properties of the matrices are obtained. Implications of the results to representation theory, and quantum information science are discussed.

Commutativity of spatiochromatic covariance matrices in natural image statistics

Statistic of natural images is a growing field of research both in vision and image processing. On the vision research side, fine statistical details about object distribution in real-world scenes help understanding the human visual system behavior. On the image processing side, by using the information gathered from statistics of natural scenes, we can obtain reliable priors and insights that can be used in many models. In has been rigorously proven in [ 16 ] that, if second order stationarity and commutativity of spatiochromatic covariance matrices hold true for natural scenes, then the codification of spatial and chromatic information by the human visual system can be separated through a tensor product. Spatial features are coded via local and oriented Fourier basis elements, while color features are coded via a triad given by an achromatic channel followed by two color opponent channels. In this paper, we will show that, while stationarity is guaranteed, commutativity is not. However, we shall see that commutativity of spatiochromatic covariance matrices can be approached if the database of images used to model visual scenes is modified accordingly to a suitable transformation that describes the response of retinal photoreceptors to light absorption: the Michaelis-Menten formula. A thorough investigation of the effects of a parameter of this formula will be performed and its influence on commutativity of covariance matrices will be detailed.

A Yang–Baxter equation for metaplectic ice

We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic n-fold cover of GL(r, F), where F is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding Yang-Baxter equation with that of the quantum affine Lie superalgebra U-root v((gl) over cap (1 vertical bar n)), modified by Drinfeld twisting to introduce Gauss sums. (The deformation parameter v is specialized to the inverse of the residue field cardinality.)For principal series representations of metaplectic groups, the Whittaker models are not unique. The scattering matrix for the standard intertwining operators is vector valued. For a simple reflection, it was computed by Kazhdan and Patterson, who applied it to generalized theta series. We will show that the scattering matrix on the space of Whittaker functions for a simple reflection coincides with the twisted R-matrix of the quantum group U-root v((gl) over cap (1 vertical bar n)). This is a piece of the twisted R-matrix for U-root v((gl) over cap (1 vertical bar n)), mentioned above.In the appendix (joint with Nathan Gray) we interpret values of spherical Whittaker functions on metaplectic covers of the general linear group over a nonarchimedean local field as partition functions of two different solvable lattice models. We prove the equality of these two partition functions by showing the commutativity of transfer matrices associated to different models via the Yang-Baxter equation.

A Quantum Algorithm for the Commutativity of Finite Dimensional Algebras

A quantum procedure for testing the commutativity of a finite dimensional algebra is introduced. This algorithm, based on Grover's quantum search, is shown to provide a quadratic speed-up (when the number of queries to the algebra multiplication constants is considered) over any classical algorithm (both deterministic and randomized) with equal success rate and shown to be optimal among the class of probabilistic quantum query algorithms. This algorithm can also be readily adapted to test commutativity and hermiticity of square matrices, again with quadratic speed-up. The results of the experiments carried out on a quantum computer simulator and on one of IBM's 5-qubit quantum computers are presented.

Multimodal image registration using Laplacian commutators

The fusion and combination of images from multiple modalities is important in many applications. Typically, this process consists of the alignment of the images and the combination of the complementary information. In this work, we focused on the former part and propose a multimodal image distance measure based on the commutativity of graph Laplacians. The eigenvectors of the image graph Laplacian, and thus the graph Laplacian itself, capture the intrinsic structure of the image’s modality. Using Laplacian commutativity as a criterion of image structure preservation, we adapt the problem of finding the closest commuting operators to multimodal image registration. Hence, by using the relation between simultaneous diagonalization and commutativity of matrices, we compare multimodal image structures by means of the commutativity of their graph Laplacians. In this way, we avoid spectrum reordering schemes or additional manifold alignment steps which are necessary to ensure the comparability of eigenspaces across modalities. We show on synthetic and real datasets that this approach is applicable to dense rigid and non-rigid image registration. Results demonstrated that the proposed measure is able to deal with very challenging multimodal datasets and compares favorably to normalized mutual information, a de facto similarity measure for multimodal image registration.

Heinz-type numerical radii inequalities

ABSTRACTThe main aim of this article is to present inequalities for the numerical radius of commutators of positive matrices. Some of these inequalities are analogues of known inequalities for unitarily invariant norms. In particular, variants, but weaker forms, of the well-known Heinz inequality and its generalizations are extended to the context of numerical radius.

Matrix 3-Lie superalgebras and BRST supersymmetry

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras [Formula: see text] if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of [Formula: see text] and [Formula: see text], we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

Saturation of the Tsirelson bound for the Clauser-Horne-Shimony-Holt inequality with random and free observables

Maximal violation of the CHSH-Bell inequality is usually said to be a feature of anticommuting observables. In this work we show that even random observables exhibit near-maximal violations of the CHSH-Bell inequality. To do this, we use the tools of free probability theory to analyze the commutators of large random matrices. Along the way, we introduce the notion of "free observables" which can be thought of as infinite-dimensional operators that reproduce the statistics of random matrices as their dimension tends towards infinity. We also study the fine-grained uncertainty of a sequence of free or random observables, and use this to construct a steering inequality with a large violation.

Some sharp bounds for the commutator of real matrices

We determine, for some values of p,q,r, the smallest constant Cp,q,rR such that‖XY−YX‖p≤Cp,q,rR‖X‖q‖Y‖r for all real square matrices X and Y, where ‖⋅‖p denotes the Schatten p-norm.

A note on trigonometric identities involving non-commuting matrices

An algorithm is presented for generating successive approximations to trigonometric functions of sums of non-commuting matrices. The resulting expressions involve nested commutators of the respective matrices. The procedure is shown to converge in the convergent domain of the Zassenhaus formula and can be useful in the perturbative treatment of quantum mechanical problems, where exponentials of sums of non-commuting skew-Hermitian matrices frequently appear.

Continuous maps on triangular matrices that preserve commutativity

We investigate continuous injective maps defined on the space of upper triangular matrices over a field F that preserve commutativity of matrices in both directions. We show that every such map is a composition of either an inner automorphism and a locally polynomial map or the two latter and one more automorphism of upper triangular matrices.