Showing papers on "Square matrix published in 1994"
TL;DR: In this paper, a formula for the inverse of a general tridiagonal matrix is given in terms of the principal minors of the matrix, where the principal minor is defined by the principal matrix.
166 citations
TL;DR: This paper shows how to calculate the sensitivity and elasticity of population growth rate to changes in the entries in the individual matrices B(i) making up a periodic matrix product and reveals seasonal patterns in sensitivity that are impossible to detect with sensitivity analysis based on the matrix A.
Abstract: Periodic matrix models are used to describe the effects of cyclic environmental variation, both seasonal and interannual, on population dynamics. If the environmental cycle is of length m, with matrices B(1), B(2),...., B(m) describing population growth during the m phases of the cycle, then population growth over the whole cycle is given by the product matrix A = B(m)B(m—1)...B(1). The sensitivity analysis of such models is complicated because the entries in A are complicated combinations of the entries in the matrices B(i), and thus do not correspond to easily interpreted life history parameters. In this paper we show how to calculate the sensitivity and elasticity of population growth rate to changes in the entries in the individual matrices B(i) making up a periodic matrix product. These calculations reveal seasonal patterns in sensitivity that are impossible to detect with sensitivity analysis based on the matrix A. We also show that the vital rates interact in important ways: the sensitivity to changes in a rate at one point in the cycle may depend strongly on changes in other rates at other points in the cycle.
131 citations
TL;DR: In this article, it was shown that it is useful to regard S = sign(A) as being part of a matrix sign decomposition A = SN, where N = (A2)1/2.
95 citations
TL;DR: In this paper, a fusion of variational and fixed point techniques was used to obtain strong analogues of the classical DAD problem, and their extensions appear inaccessible by either technique separately.
88 citations
TL;DR: It is suggested that the spectral properties near zero virtuality of three-dimensional QCD follow from a Hermitian random matrix model, which is a family of random matrix models for both even and odd number of fermions.
Abstract: We suggest that the spectral properties near zero virtuality of three-dimensional QCD follow from a Hermitian random matrix model. The exact spectral density is derived for this family of random matrix models for both even and odd number of fermions. New sum rules for the inverse powers of the eigenvalues of the Dirac operator are obtained. The issue of anomalies in random matrix theories is discussed.
82 citations
TL;DR: A simple method which leads to results on the Lipschitz continuity of the matrix absolute value and several new ones is discussed.
55 citations
TL;DR: In this article, it is shown that every pure Mueller matrix has a simple and elegant structure, which is embodied by interrelations that involve either only squares of the elements or only products of different elements.
Abstract: Changes in the radiance and state of polarization of a beam of radiation can often be described by means of a pure Mueller matrix. Such a 4 × 4 matrix transforms Stokes parameters and can be expressed in terms of the elements of a 2 × 2 Jones matrix. Relations between the two types of matrix are discussed. Explicit expressions are given for changes of a pure Mueller matrix that are caused by certain elementary changes of its Jones matrix, such as when its transpose, complex conjugate, or Hermitian conjugate are taken. It is shown that every pure Mueller matrix has a simple and elegant structure, which is embodied by interrelations that involve either only squares of the elements or only products of different elements. All possible interrelations for the elements of a general pure Mueller matrix are derived from this simple structure.
55 citations
TL;DR: In this paper, the statistical properties of matrix elements of Hermitian operators in an eigenbasis of a typical bounded Hamiltonian, which can be classically either ergodic, or integrable, or somewhere in between (mixed dynamics), are addressed.
40 citations
TL;DR: In this article, the problem of finding a nonnegative symmetric matrix with row maxima prescribed by a positive vector is studied and an algorithm for finding such a matrix is described.
Abstract: A nonnegative symmetric matrix $B$ has row maxima prescribed by a given vector $r$, if for each index $i$, the maximum entry in the $i$th row of $B$ equals $r_i$. This paper presents necessary and sufficient conditions so that for a given nonnegative symmetric matrix $A$ and positive vector $r$ there exists a positive diagonal matrix $D$ such that $B = DAD$ has row maxima prescribed by $r$. Further, an algorithm is described that either finds such a matrix $D$ or shows that no such matrix exists. The algorithm requires $O(n \lg n + p)$ comparisons, $O(p)$ multiplications and divisions, and $O(q)$ square root calculations where $n$ is the order of the matrix, $p$ is the number of its nonzero elements, and $q$ is the number of its nonzero diagonal elements. The solvability conditions are compared and contrasted with known solvability conditions for the analogous problem with respect to row sums. The results are applied to solve the problem of determining for a given nonnegative rectangular matrix $A$ positive, diagonal matrices $D$ and $E$ such that $DAE$ has prescribed row and column maxima. The paper presents an equivalent graph formulation of the problem. The results are compared to analogous results for scaling a nonnegative matrix to have prescribed row and column sums and are extended to the problem of determining a matrix whose rows have prescribed $l_p$ norms.
36 citations
TL;DR: The circular fuzzy matrices are defined and some properties of a square fuzzy matrix (such as reeflexivity, transitivity, circularity) are carried over to the adjoint of the matrix.
TL;DR: In this article, the authors obtained explicit formulas for the coefficients of a second order difference block operator if its spectral or its scattering functions are rational matrix functions analytic and invertible on the unit circle.
Abstract: In this paper we obtain explicit formulas for the coefficients of a second order difference block operator if its spectral or its scattering functions are rational matrix functions analytic and invertible on the unit circle. The solutions are given in terms of realizations of the spectral or scattering function.
TL;DR: It is shown that this minimum scaling of an interval matrix with a stable center matrix is a MAX-SNP-hard problem and cannot be approximated with a ratio arbitrarily close to unity in polynomial time.
Abstract: An interval matrix can be represented in terms of a “center” matrix and a nonnegative error matrix, specifying maximum elementwise perturbations from the center matrix. A commonly proposed robust stability (regularity) characterization for an interval matrix with a stable (nonsingular) center matrix identifies the minimum scaling of this error matrix for which instability (singularity) is achieved. In this paper it is shown that approximating this minimum scaling is a MAX-SNP-hard problem. This implies that in the general case, unless the class of deterministic polynomial-time decision problems, P, equals the class of nondeterministic polynomial-time decision problems, NP, thought to be highly unlikely, this minimum scaling cannot be approximated with a ratio arbitrarily close to unity in polynomial time.
TL;DR: In this article, a matrix equation, containing the covariance matrix, is derived, and solved for the MA, AR, and ARMA cases, and the result is quite, and maybe surprisingly, simple.
TL;DR: Bounds for the singular values, ratios of singular values and rank of a square matrix A, involving tr A, tr A 2, and tr A H A, are presented in this paper.
Patent•
13 Oct 1994TL;DR: In this paper, the authors proposed a simulation method for obtaining internal information at discrete points within a semiconductor device to be analyzed based on information related to the semiconductor devices by forming a coefficient matrix A having matrix equation A·X=b with respect to each of the discrete points and solving the matrix equation, where X denotes a matrix of physical quantities to be obtained and b is a known matrix.
Abstract: A simulation method for obtaining internal information at discrete points within a semiconductor device to be analyzed based on information related to the semiconductor device by forming a coefficient matrix A having matrix equation A·X=b with respect to each of the discrete points within the semiconductor device and solving the matrix equation. X denotes a matrix of physical quantities to be obtained and b is a known matrix. The device simulation method includes the steps of (a) assigning a rectangular work region to the semiconductor device regardless of its shape so that it fits within the work region, (b) forming a matrix of elements with respect to each of the discrete points within the analyzing region and dummy elements with respect to each of the discrete points within a non-analyzing region at least within the work region. The dummy element have values substituted into the matrix such that they form a diagonal term of the matrix respectively with a positive integer and remaining dummy elements respectively are "0", (c) forming a matrix (L·U) by dividing the matrix into lower and upper triangular matrices L and U, (d) calculations using a stored iteration method and the specific matrix (L·U) until a convergence condition is satisfied, to obtain a solution for each element of the matrix X, and (e) determining, using the solution for each element of the matrix X, whether the semiconductor device is operational.
TL;DR: The classical Rudin-Shapiro construction as discussed by the authors produces a sequence of polynomials with ± 1 coefficients such that on the unit circle each such polynomial P satisfies the "flatness" property ||P||∞ ≤ √2||P||2.
Abstract: The classical Rudin–Shapiro construction produces a sequence of polynomials with ±1 coefficients such that on the unit circle each such polynomial P satisfies the "flatness" property ||P||∞ ≤ √2||P||2. It is shown how to construct blocks of such flat polynomials so that the polynomials in each block form an orthogonal system. The construction depends on a fundamental generating matrix and a recursion rule. When the generating matrix is a multiple of a unitary matrix, the flatness, orthogonality, and other symmetries are obtained. Two different recursion rules are examined in detail and are shown to generate the same blocks of polynomials although with permuted orders. When the generating matrix is the Fourier matrix, closed-form formulas for the polynomial coefficients are obtained. The connection with the Hadamard matrix is also discussed.
19 Apr 1994
TL;DR: A new technique is proposed for estimating the mutual coupling matrix of an array of general geometry, which employs one pilot source, and uses an extra element at some distance away from the original array.
Abstract: The adverse effect of mutual coupling between the elements of an array, on the performance of super-resolution techniques is demonstrated. An analytical modelling of the mutual coupling is presented, and its effects are modelled in the form of a complex square matrix, the mutual coupling matrix (MCM). The MCM depends on the array geometry and the array electrical characteristics, but not on the direction of the incoming signals. Based on the modelling of the mutual coupling effects, a new technique is proposed for estimating the mutual coupling matrix of an array of general geometry. The proposed method employs one pilot source, and uses an extra element at some distance away from the original array. >
TL;DR: A numerical method for the exact reduction of a singlevariable polynomial matrix to its Smith form without finding roots and without applying unimodular transformations is presented.
Abstract: In the present paper is presented a numerical method for the exact reduction of a singlevariable polynomial matrix to its Smith form without finding roots and without applying unimodular transformations. Using the notion of compound matrices, the Smith canonical form of a polynomial matrixM(s)∈ℝnxn[s] is calculated directly from its definition, requiring only the construction of all thep-compound matricesCp(M(s)) ofM(s), 1
TL;DR: In this paper, a graph-theoretic procedure is derived to reveal a possibly hidden factorizability of the determinant det( sE − A ), which is very important for large-scale systems.
TL;DR: In this paper, a general computing strategy to compute Kappa coefficients using the SPSS MATRIX routine is proposed, which is based on the following rationale: if the contingency table is considered as a square matrix, then the observed proportions of agreement lie in the main diagonal's cells, and their sum equals the trace of the matrix, whereas the proportions of expected by chance are the joint product of marginals.
Abstract: This short paper proposes a general computing strategy to compute Kappa coefficients using the SPSS MATRIX routine. The method is based on the following rationale. If the contingency table is considered as a square matrix, then the observed proportions of agreement lie in the main diagonal’s cells, and their sum equals the trace of the matrix, whereas the proportions of agreement expected by chance are the joint product of marginals. The generalization to weighted kappa, which requires an additional square matrix of disagreement weights, both matrices having the same order, becomes possible by the use of the Hadamard product-that is, the elementwise direct product of two matrices.
TL;DR: This paper introduces two new classes of L -matrices, which for square matrices reduce to sign-nonsingular matrices, and the maximum number of columns for matrices in each of these classes is obtained, and those matrices attaining the maximum are characterized.
TL;DR: In this paper, an alternative limit expression of the Drazin inverse of a square matrix is presented, and a new proof for a finite algorithm for determining the inverse is given using the new limit expression.
Patent•
13 Sep 1994TL;DR: In this article, the authors proposed a method for collective evaluation of the old spectra by the following steps: a) formation of a data matrix from the spectral vectors formed by the series of spectral values of a respective spectrum, the spectral vector being arranged in the columns (or in the rows) of the data matrix in a location-dependent manner, b) singular value decomposition of the dataset matrix in order to obtain three matrices whose product corresponds to the data matrices, the first matrix consisting of spectrally dependent vectors, the second matrix is a diagonal matrix and
Abstract: The invention relates to an examination method whereby a respective spectrum with a number of spectral values is measured for a number of locations. Collective evaluation of the old spectra is enabled by the following steps: a) formation of a data matrix from the spectral vectors formed by the series of spectral values of a respective spectrum, the spectral vectors being arranged in the columns (or in the rows) of the data matrix in a location-dependent manner, b) singular value decomposition of the data matrix in order to obtain three matrices whose product corresponds to the data matrix, the first (third) matrix consisting of spectrally dependent vectors, whereas the second matrix is a diagonal matrix and the third (first) matrix consists of location-dependent vectors, c) evaluation of at least one of the three matrices.
TL;DR: The coefficients of the characteristic polynomial of a matrix are expressed solely as functions of the traces of the powers of the matrix and some combinatorial identities are derived.
TL;DR: In this article, an iterative scheme of first order is developed for the purpose of solving linear systems of equations, in particular systems that are derived from boundary integral equations are investigated, and the iterative schemes to be considered are of the form Ex(k+1) = Dx(k) + d, where E and D are square matrices.
Abstract: In this paper an iterative scheme of first order is developed for the purpose of solving linear systems of equations. In particular, systems that are derived from boundary integral equations are investigated. The iterative schemes to be considered are of the form Ex(k+1) = Dx(k) + d, where E and D are square matrices. It will be assumed that E is a lower matrix, i.e. the coefficients above the central diagonal are zero. It will be shown that by considering matrix D embedded in a vector space and reducing its size with respect to a chosen metric, that convergence rates can be substantially improved. Equation ordering and parameter matrices are used to reduce the magnitude of D. A number of examples are tested to illustrate the importance of the choice of metric, equation ordering and the parameter matrix. Computation times are determined for both the iterative procedure and Gauss elimination indicating the usefulness of iteration which can be orders of magnitude faster.
TL;DR: In this article, the authors construct limit representations of weighted pseudoinverse matrices with singular weights, which are represented in terms of the coefficients of characteristic polynomials of certain square matrices.
Abstract: Weighted pseudoinverse matrices with singular weights are represented in terms of the coefficients of characteristic polynomials of certain square matrices. By using the expression obtained, we construct limit representations of weighted pseudoinverse matrices with singular weights.
TL;DR: This characterization is used to determine the sign patterns of the inverses of fully indecomposable, strong sign-nonsingular matrices and to develop a recognition algorithm for such sign patterns.
TL;DR: In this article, the authors studied the asymptotic behavior of sequences of non-singular square matrices whose terms are block-diagonal (diagonal, respectively) matrices plus some perturbation term.
TL;DR: In this paper, the transfer function matrix of a two-dimensional system is computed in terms of the original system matrix and does not require the inversion of a 2-variable polynomial matrix.
Abstract: An algorithm is developed for the computation of the transfer function matrix of a two-dimensional system, which is given in its generalized form The algorithm is a recursion in terms of the original system matrix and does not require the inversion of a two-variable polynomial matrix An algorithm for the evaluation of the Laurent expansion of the inverse of a two-variable polynomial matrix is also presented