Showing papers on "Square matrix published in 1979"

The Commutation Matrix: Some Properties and Applications

Abstract: The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

The commutation matrix: Some properties and applications

Abstract: The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

Vec and vech operators for matrices, with some uses in jacobians and multivariate statistics

Abstract: The vec of a matrix X stacks columns of X one under another in a single column; the vech of a square matrix X does the same thing but starting each column at its diagonal element. The Jacobian of a one-to-one transformation X → Y is then ∣∣∂(vecX)/∂(vecY) ∣∣ when X and Y each have functionally independent elements; it is ∣∣ ∂(vechX)/∂(vechY) ∣∣ when X and Y are symmetric; and there is a general form for when X and Y are other patterned matrices. Kronecker product properties of vec(ABC) permit easy evaluation of this determinant in many cases. The vec and vech operators are also very convenient in developing results in multivariate statistics.

On the independence theory of equalizer convergence

Abstract: High-speed pulse amplitude modulated (pam) data transmission over telephone channels is only possible when adaptive equalization is used to mitigate the linear distortion found on the (initially unknown) channel. At the beginning of the equalization procedure, the tap weights are adjusted to minimize the inter symbol interference between pulses. The “stochastic gradient” algorithm is an iterative procedure commonly used for setting the coefficients in these and other adaptive filters, but a proper understanding of the convergence has never been obtained. It has been common analytical practice to invoke an assumption stating that a certain sequence of random vectors which direct the “hunting” of the equalizer are statistically independent. Everyone acknowledges this assumption to be far from true, just as everyone agrees that the final predictions made using it are in excellent agreement with experiments and simulations. We take the resolution of this question as our main problem When one begins to analyze the performance of the algorithm, one sees that the average mean-square error after the nth iteration requires knowing, as an intermediate step, the mathematical expectation of the product of a sequence of statistically dependent matrices. We transform the latter problem to a space of sufficiently high dimension where the required average may be obtained from a canonical equation V n+1 = A(α)V n + Here A(α) is a square matrix, depending on the “step-size” α of the original algorithm, and V n and F are vectors. The mean-square error is calculable from the solution V n .

Chapter 2 – nonnegative matrices

Abstract: Publisher Summary This chapter discusses square nonnegative matrices, that is, square matrices all of whose elements are nonnegative. The matrices A that satisfy A > 0 are called positive matrices. The two basic approaches to the study of nonnegative matrices are geometrical and combinatorial. The chapter describes the combinatorial matrices, using the elementwise structure in which the zero–nonzero pattern plays an important role. It also discusses the irreducible matrices and the reducible case. An irreducible matrix is primitive if its trace is positive. The order of cyclicity of a matrix can be computed by inspection of its directed graph. If there are n possible states of a certain process and the probability of the process moving from state si to state sj is time independent, such a process is called a finite homogeneous Markov chain. The maximal eigenvalue of a stochastic matrix is one. A nonnegative matrix T is stochastic if and only if e is an eigenvector of T corresponding to the eigenvalue one.

Stability for a Multi-Rate Sampled-Data System

Abstract: A sampled-data system with sampling interval lengths selected from a finite set is considered. Stabilizability of the system via feedbacks associated with sampling interval lengths is studied, and conditions for stabilizability involving “pre-contractiveness”, “contractiveness” and “positive definiteness” of a finite set of matrices are given. Included in these results is a generalization of a theorem by P. Stein stating that for a real square matrix H, $\lim _{n \to \infty } H^n = 0$ if and only if there is a symmetric matrix Q such that $Q - H^T QH$ is positive definite. Finally, some results concerning a choice of feedbacks which will produce stability are presented.

The cumulative distance from an observed to a stable age structure.

Abstract: In a closed, unisexual, age-structured population with age-specific birth and death rates which are constant in time, the vector describing a census by age categories will, as time increases, approach proportionality to the stable age structure implied by the vital rates. This stable age structure is the dominant eigenvector of a demographic projection matrix which carries out the action on a census vector of the age-specific vital rates. We show that the elementwise convergence to zero of the discounted deviations from the stable age structure is complete and exponential. The sum, over all time, of the signed discounted deviations may be easily calculated from a fundamental matrix based on the projection matrix. These results are proved for any primitive nonnegative square matrix. In the demographic context, these results suggest alternatives to an index which has been used to measure the distance from an observed to a stable age structure.

Unsolved problems in matrix and operator theory, II. partial multiplicities for products

Abstract: The purpose of this note is to pose a problem about the partial multiplicities of a product of two matrix polynomials given those of its factors.

Cones, graphs and optimal scalings of matrices †

Abstract: Characterizations are given of the optimal scalings of a complex square matrix within its diagonal similarity class and its restricted diagonal equivalence class with respect to the maximum element norm. The characterizations are in terms of a finite number of products, principally circuit and diagonal products. The proofs proceed by reducing the optimal scaling problems from the multiplicative matrix level in succession to an additive matrix level, a graph theoretic level, and a geometric level involving duality theorems for cones. At the geometric level, the diagonal similarity and the restricted diagonal equivalence problems are unified.

Recognition of angular orientation of objects with the help of optical sensors

Abstract: When using photodiode arrays for optical recognition, the information of a picture is represented in a fixed orthogonal grid. This grid corresponds to the matrix arrangement of the individual photodiodes. In a square matrix of n × n points, the information in the picture can be presented as a sequence of light intensity values according to equation 1:

A new method of matrix transformation. I. Matrix diagonalizations via involutional transformations

Abstract: It is shown that two matrices A and B of order n×n which satisfy a monic quadratic equation with two roots λ1 and λ2 are connected by ATAB=TABB where TAB=A+B−(λ1+λ2) I with I being the n×n unit matrix (Theorem 1). The condition for TAB to be involutional is that the anticommutator of ?=A−(1/2)(λ1+λ2) I and ?=B−(1/2)(λ1+λ2) is a c number (Theorem 2). A 2m×2m matrix Q(2m) is introduced as a typical form of a matrix which can be diagonalized by an involutional transformation. These theorems are further extended through the matrix representation of the group of the general homogeneous linear transformations, GL(n). IUH (involutional, unitary, and Hermitian) matrices are introduced and discussed. The involutional transformations are shown to play a fundamental role in the transformations of Dirac’s Hamiltonian and of the field Hamiltonians which are quadratic in particle creation and annihilation operators in solid state physics.

A new method of matrix transformations. II. General theory of matrix diagonalizations via reduced characteristic equations and its application to angular momentum coupling

Abstract: A general formalism is given to construct a transformation matrix which connects two matrices A and B of order n×n satisfying any given polynomial equation of degree r, p(r)(x) =0; r?n. The transformation matrix TAB is explicitly given by a polynomial of degree (r−1) in A and B based on p(r)(x). A special case where B is a diagonal matrix Λ equivalent to A leads to the general theory of matrix diagonalizations with the transformation matrix TAΛ, which can be made nonsingular with a proper choice of Λ. In another special case where B is a constant matrix with the constant being a simple root λν of p(r)(x), the transformation matrix TAB reduces to the idempotent matrix Pν belonging to the eigenvalue λν of A. Based on the relation which exists between TAΛ and Pν, one can construct a transformation matrix U which is more effective than TAΛ and becomes unitary when A is Hermitian. Illustrative examples of the formalism are given for the problem of angular momentum coupling.

Linear time-varying systemsߞstate transition matrix

Abstract: The state transition matrix of a linear time-varying system cannot, in general, be expressed in a closed form and has, therefore, to be evaluated numerically. For the commutative class of linear time-varying systems, the state-transition matrix is the exponential matrix. A numerical procedure is developed for the evaluation of this matrix to any desired degree of accuracy by the method of series expansion. For linear time-varying systems, which are not restricted to belonging to the commutative class, an efficient computational algorithm is developed for the evaluation of the state-transition matrix. This is based on the minimum m.s.e. approximation of a time function in terms of a set of block-pulse functions, which are orthogonal in the speficied interval. The algorithms developed in the paper are illustrated by appropriate examples.

Constructions of modular forms by means of transformation formulas for theta series

Abstract: where U, V are mXn real matrices, tr denotes the trace of a corresponding square matrix and G runs through allmXn integral matrices. We write simply 0F,u,v(Z) for the theta series 0f,u,v(Z;0) when 0 is of order 0. For congruence subgroups of SL^{Z) the transformation formulas for theta series of degree 1 associated with F are well known. For example, we can find transformation formulas for theta series of degree 1 in [7],[8],in which multipliers are explicitly determined. Transformation formulas for the theta series 0f,u,v{Z',O) of degree n>l are also established in [1] in the case where F is even and U, V are zero (the condition on U, V is not necessary if 0 is of order 0 [9]). Using these results we can get many examples of Siegel modular forms for congruence subgroups. In this paper we determine a transformation formula for the theta series 0f,u,v(Z;0) associated with a positive integral symmetric matrix F and any real matrices U, V and using this, we get some examples of cusp forms for some congruence subgroups F' of Spn(Z). Cusp forms of weight n + 1 for Fr induce differentialforms of the first kind on the nonsingular model of the modular function fieldwith respect to /''. Our result shows that the geometric genus of the nonsingular model of the modular function fieldwith respect to F' is positive.

The inverse of a certain block matrix

Abstract: A simple formula for the inverse of a block matrix with non-zero blocks in the principal diagonal and the first sub-diagonal only is proved. The matrix had arisen in an investigation of a difference equation.

An Alternate Approach to Transitive Coupling in ISM

Abstract: Transitive coupling is a problem encountered in structuring complex systems using the interpretive structural modeling (ISM) process. It involves the interconnection of two systems defined on the same contextual relation that is transitive. A new approach to transitive coupling, which does not require the development of the implication matrix, is presented. The main drawback in using the implication matrix is the considerably large amount of computer storage required by this matrix. Since the computer is used interactively in the ISM process, the amount of storage available for use is usually limited. The new method surmounts this difficulty by using only square matrices of size equal to the total number of elements involved in transitive coupling. The new approach leads to an alternative procedure for transitive bordering as well.

Berechnung der eigenwerte einer matrix mit dem hr-verfahren

Abstract: The HR-process, a method of computing the eigenvalues of a real square matrix, is presented. It leaves the pseudosymmetric form of a matrix invariant and may be considered as a generalization of the well-known QR-process. Computations are much faster with the HR-process than with the QR-process for the case of large non-symmetric matrices of tridiagonal form and for matrices which can be reduced to such a form in a well-behaved manner.

S-matrix theory and the density matrix formalism

Abstract: A general equation governing the time development of the diagonal part of the density matrix is proposed for weakly interacting systems possessing no off-diagonal long range order. This equation, which involves generalized Moller operators of the type employed in S-matrix theory, is solved for two cases and leads to the generalized Pauli and Boltzmann equations.

A variational principle for the linear filter matrix and an interpretation for the maximum value of the functional

Abstract: A functional of a trial matrix and suitable correlation matrix is constructed which is an absolute maximum when the trial matrix satisfies the Wiener-Hopf equation for the filter matrix. The maximum value of the functional is essentially the sum of the squares of the minimum errors of the observations and is thus of interest in its own right.