Commutative Matrices Research Articles

On the Parametrization of a Reachable Set of a Bilinear System with Commuting Matrices

On the Parametrization of a Reachable Set of a Bilinear System with Commuting Matrices

On second-order and fourth-order moments of jointly distributed random matrices: a survey

The study is concerned with second-order and fourth-order moments of jointly distributed random matrices. When distributional properties are required, normality is adopted. Some of the results can also be applied to elliptical or Wishart distributions. The developments are entirely algebraic. Full use is made of the Kronecker product, (repeated) vectorization, commutation matrices and related items. There are a few references in the main text but many additional references to other work, in which the same or kindred results (obtained by other methods, procedures or concepts) can be found, have been included in the References.

On the structure of commutative matrices. II

A finite set A of N × N nilpotent commutative matrices that have one-dimensional joint kernel is considered. The theorem (due to Suprunenko and Tyshkevich) that the algebra A generated by A and the identity matrix has dimension equal to N is proved. A canonical basis for A is given, and related structure constants are discussed.

On the structure of commutative matrices. II

A finite set A of N × N nilpotent commutative matrices that have one-dimensional joint kernel is considered. The theorem (due to Suprunenko and Tyshkevich) that the algebra A generated by A and the identity matrix has dimension equal to N is proved. A canonical basis for A is given, and related structure constants are discussed.

Reachable sets and controllability of bilinear time-invariant systems: a qualitative approach

This paper is concerned with a qualitative method for the study of reachable sets and controllability of bilinear time-invariant systems. A nonlinear decomposition technique which reduces the solution of the problem to the analysis of the qualitative behavior of a finite number of linear uncontrolled systems is introduced. Special consideration is given to the homogeneous bilinear systems with commutative matrices. Two- and three-dimensional examples are discussed.

Structure of algebras of commutative matrices

Commutative spaces of matrices A = ∑ n k=1 a k J k are studied, where { J k } is a set of (0, 1) matrices with prescribed sum S, and the a k 's are complex parameters. The great number of commutative spaces A obeying simple prescriptions on the first two rows of S can all be described in terms of two basic algebras of matrices denoted by T n and Γ n . A relationship between the structure of spaces A and the multiplicative complexity of the bilinear forms defined by the J k 's is discussed.

On the structure of commutative matrices

A finite set of commutative matrices is viewed as a cubic array. Its structure is considered via a collection of related symmetric matrices.

Third moment of matrix quadratic form

The third moment of a matrix quadratic form is obtained using the sixth moment of the matrix normal distribution and applying the properties of commutation matrices and Kronecker products.

Higher order moments of multivariate normal distribution using matrix derivatives

A general formula for the central moments of multivariate normal distribution is derived by differentiating its characteristic function using matrix derivatives. An explicit expression for the moments is obtained. Two applications of these results are given. The sixth order moments are arranged in a square matrix using the properties of commutation matrices and vec operators

On linear ordinary differential equations with functionally commutative coefficient matrices

A system of linear ordinary differential equations x ̇ = A (t) x , t ∈ I , is called semiproper if A (t) A (τ)= A (τ) A (t) , t , τ ∈ I [also known as functional commutativity of A (t) ]. It is known that a semiproper system has a closed-form fundamental solution matrix X A (t)=exp∫ t A (τ)dτ , where the matrix exponential is defined by the power series exp(·)=Σ ∞ k=1 (·) k k! . Therefore the problem of solving a semiproper system amounts to that of finding a finite-form expression for the matrix exponential. Based on some recent results obtained by the authors for decomposing semiproper matrix functions, a systematic approach is developed for finding a finite-form analytical solution for the entire family of semiproper systems. This solution is then used to derive a number of important and practical stability criteria for semiproper systems. Applications of the new results are also discussed.

The classical parafermion algebra, its generalization and its quantization

The Poisson bracket algebra of the classical parafermions derived earlier from the lagrangian description of conformal coset models is generalized. It is also shown how to quantize models with commutative monodromy matrices, and progress is made in quantizing the non-commutative case.

Numerical solution of joint eigenpairs of a family of commutative matrices

A numerical method for solving joint eigenpairs of a family of commuting matrices is presented. The stability of algorithm and perturbation bounds of the problems are discussed.

Spatial decomposition of functionally commutative matrices

Let I ⊆ R and D ⊆ C , where R and C are the fields of real and complex numbers, respectively. Let C n×n be the space of square matrices of order n over C . A matrix-valued function F : I → C n×n is said to be proper on I if F ( t ) = f ( t , A ), where A ϵ C n×n and f : I × D → C is a scalar function, and F is said to be semiproper on I if F ( t ) F ( t ) = F ( τ ) F ( t ) for all t , τ ϵ I . The main results presented here are: (1) a characterization together with some potentially useful results for proper matrix functions; (2) a characterization of semiproper matrix functions in terms of proper ones; (3) a new and systematic procedure for decomposing a semiproper matrix function into a finite sum of mutually commutative proper ones. Some important applications of these new results in control engineering, linear systems theory, and the theory of linear differential equations are also included.

2D systems feedback compensation: an approach based on commutative linear transformations

Algebraic properties of a pair of commutative matrices associated with an ideal in R[z1, z2] are exploited for characterizing the closed loop polynomial variety of a 2D system. Also algorithms are given to find under what constraints the closed loop variety can be assigned and to compute the MFD of a compensator.

Higher order moments of random vectors using matrix derivatives

Matrix derivative methods and properties of commutation matrices are used to obtain higher order moments of a random vector from its characteristic function. The method is illustrated on the multivariate normal distribution

Matrices With Applications in Statistics

Prerequisite matrix theory. Prerequisite vector theory. Linear transformations and characteristic roots. Geometric interpretations. Algebra of vector spaces. Generalized inverse. Conditional inverse. Systems of linear equations. Patterned matrices and certain other special matrices. Trace of a matrix, vector of a matrix, commutation matrices. Integration and differentiation. Positive matrices and matrices with non-positive off-diagonal elements. Non-negative matrices, idempotent and tripotent matrices, projections.

An application of the Cayley-Hamilton theorem to matrix polynomials in several variables

Let Σ={m 1}|Σ|2 be a finite set of square n × n matrices. Let Lp be the linear space with rank rp (n 2) spanned by the set of matrices (m:meσ∗,|m| p and let c(σ) be defined by c(σ) = min {p:rp =r p+1, where |X| denotes the number of elements in a set σ or the length of a word σ and σ denotes the set of all finite words over σ (here a word is a sequence of matrices from σ and is understood as the matrix product of the matrices in the sequence). It is shown in the paper that C(σ) L(n 2+2)/31. The proof of the above bound is based on combinatorial arguments and it is not known whether the bound is sharp. It follows from the Cayley-Hamiiton Theorem that c(σ)n−1if σ a singleton. It is also shown (a result of H. W. Lenstra Jr.) that the bound C(σ)n−1 holds for commutative matrices.

Extrema of quadratic forms and statistical applications

Khatri and Rao gave a bound to the ratio of the determinants . Where X is an n x k matrix of rank k, and B, C are commutative matrices. In this paper, we provide bounds to the ratios of the determinants and a general result then that of Khatri and Rao. The results can be applied to some problems In statistics

Some results on commutation matrices, with statistical applications

AbstractThe commutation matrix Pmn changes the order of multiplication of a Kronecker matrix product. The vec operator stacks columns of a matrix one under another in a single column. It is possible to express the vec of a Kronecker matrix product in terms of a Kronecker product of vecs of matrices. The commutation matrix plays an important role here. “Super‐vec‐operators” like vec A ⊗ vec A vec (A ⊗ A), and vec{(A ⊗ A)Pnn} are very convenient. Several of their properties are being studied. Both the traditional commutation matrix and vec operator and the newer concepts developed from these are applied to multivariate statistical and related problems.

Multiple-parameter tunable digital filters using submultiple sampling techniques

A multiple-parameter variable digital filter is proposed which is derived from periodically switched analogue networks. A bilinear expansion of the system matrix with pairwise commutative matrices leads to a structure in which the significant parameters can be tuned independently. This method is demonstrated by a 2nd-order digital filter section. As an example, the results of a 2-parameter tunable 3rd-order lowpass filter are presented.