Showing papers on "Square matrix published in 1991"
TL;DR: In this article, a unitary transformation method that transforms the complex covariance matrix of an equally spaced linear array, which is Hermitian persymmetric, and the complex search vector into a real symmetric matrix and a real vector, respectively, is presented.
Abstract: Eigenstructure methods for estimating angles of arrival of radiation sources generally require complex computations in computing eigencomponents of the covariance matrix and calculating the search function. A unitary transformation method that transforms the complex covariance matrix of an equally spaced linear array, which is Hermitian persymmetric, and the complex search vector into a real symmetric matrix and a real vector, respectively is presented. Both tasks can be accomplished by real computations. The sampled covariance matrix available is not persymmetric. To suit the unitary transformation method, a persymmetrized estimator of the sampled covariance matrix, which is optimal in the sense of Euclidean distance, is proposed. >
161 citations
TL;DR: In this article, the integrable Calogero type and its generalizations can be obtained from gauged one-dimensional matrix models including a fermionic part and a Chern-Simons term.
74 citations
TL;DR: A new definition of determinant over the quaternion field is given in this paper, where Cramer solutions of right (or left) linear equations and the condition of existence of inverse square matrices are obtained from the definition.
Abstract: A new definition of determinant over the quaternion field is given in this paper Cramer solutions of right (or left) linear equations and the condition of existence of inverse square matrices are obtained from the definition.
68 citations
TL;DR: In this paper, the integrable systems that arise in the continuum theory are shown to result directly from the formulation of the matrix integrals in terms of orthogonal polynomials.
Abstract: The matrix integrals involved in 2d lattice gravity are studied at finiteN. The integrable systems that arise in the continuum theory are shown to result directly from the formulation of the matrix integrals in terms of orthogonal polynomials. The partition function proves to be a tau function of the Toda lattice hierarchy. The associated linear problem is equivalent to finding the polynomial basis which diagonalizes the partition function. The cases of one Hermitian matrix, one unitary matrix, and Hermitian matrix chains all fall within the Toda framework.
66 citations
TL;DR: In this paper, a method for computing the inverse of a complex n-block tridiagonal quasi-Hermitian matrix using an adequate number of partitions of the complete matrix is presented.
Abstract: This paper presents a method for computing the inverse of a complex n-block tridiagonal quasi-Hermitian matrix using an adequate number of partitions of the complete matrix. This type of matrix is very usual in quantum mechanics and, more specifically, in solid state physics (e.g. interfaces and super-lattices), when the tight-binding approximation is used. The efficiency of the method is analysed by comparing the required CPU time and work-area with other techniques.
53 citations
TL;DR: In this paper, a new class of invariant minimax estimators for the case p > m + 1, which are multivariate extensions of the estimators of Stein and Baranchik, is proposed.
46 citations
TL;DR: The inner-outer factorization for square real rational matrices which may have zeros on the jω-axis including infinity is discussed and a factorization method is given in terms of the descriptor form representation of the rational matrix.
45 citations
TL;DR: In this paper, the ordering of two Hermitian nonnegative definite matrices A and B relates to their ordering of their squares A 2 and B 2, in the sense of the Lowner partial ordering, the minus partial ordering and the star partial ordering.
35 citations
TL;DR: An enhanced version of a signal is obtained after subtraction of a linear combination of rank one matrices from the signal observation matrix by averaging the elements of this matrix, and a reconstructed signal is generated.
Abstract: An enhanced version of a signal is obtained after subtraction of a linear combination of rank one matrices from the signal observation matrix. This operation results in a nonToeplitz observation matrix. By averaging the elements of this matrix, a new Toeplitz matrix is produced and a reconstructed signal is generated. This averaging operation is examined, and physical interpretations are given. >
35 citations
TL;DR: In this article, the authors consider a real square matrix A of order n which satisfies A+At = J−I (where J is the all ones matrix) and its score vectors= Al (where 1 is a all ones vector).
Abstract: Consider a real square matrix A of order n which satisfies A+ At = J− I (where J is the all ones matrix) and its score vectors= Al (where 1 is the all ones vector). Here are our main results. If , then A has a real positive eigenvalue p with while the other eigenvalues satisfy A has eigenvalues p and λ such that and if and only if A has n − 2 eigenvalues with real part -1/2. If then for any eigenvalue . Further, if then a Perron-Frobenius result holds for A. A consequence of this is that if such an A is non-negative as well, and if n ≥ 9, then A is irreducible and primitive.
TL;DR: In this article, the authors show that the group inverse of a square matrix A of rank r over an integral domain R has a group inverse if and only if the sum of all r × r principal minors of A is an invertible element of R.
CERN1
TL;DR: In this paper, the problem of reconstructing a unitary matrix from the knowledge of the moduli of its matrix elements was studied for both symmetric and non-symmetric matrices.
Abstract: We study the problem of reconstructing a unitary matrix from the knowledge of the moduli of its matrix elements, first in the case of a symmetric matrix, which could be theS matrix forn coupled channels, second in the case of a non-symmetric matrix like the Cabibbo-Kobayashi-Maskawa matrix forn generations of quarks and leptons. In the symmetric case we find conditions under which the problem has\(2^{(n^2 - 3n)/2} \) solutions differing in a non-trivial way, but also situations where one has continuous ambiguities.
TL;DR: In this paper, the following problem was solved: under what conditions does there exist a square matrix A = [A ij ], i, j ∈ {1,2} (A 11 is principal) over an arbitrary field, with prescribed eigenvalues and prescribed submatrices A 11, A 12, and A 22?
TL;DR: In this paper, the problem of finding a mapping 2' assigning to each polynomial f of degree n a vector X(f) E C such that the matrix B := A - a is a companion matrix off is investigated.
TL;DR: The proposed algorithm is a modification of the well known algorithm due to Rust et al.
Abstract: The generalized inverse of a matrix is an extension of the ordinary square matrix inverse which applies to any matrix (e.g., singular, rectangular). The generalized inverse has numerous important applications such as regression analysis, filtering, optimization and, more recently, linear associative memories. In this latter application known as Distributed Associative Memory, stimulus vectors are associated with response vectors and the result of many associations is spread over the entire memory matrix, which is calculated as the generalized inverse. Addition/deletion of new associations requires recalculation of the generalized inverse, which becomes computationally costly for large systems. A better solution is to calculate the generalized inverse recursively. The proposed algorithm is a modification of the well known algorithm due to Rust et al. [2], originally introduced for nonrecursive computation. We compare our algorithm with Greville's recursive algorithm and conclude that our algorithm provides better numerical stability at the expense of little extra computation time and additional storage.
TL;DR: Two iterative techniques to compute an inverse square root of a given matrix are suggested and their numerical stability properties are investigated.
Abstract: The inverse square root of a matrix plays a role in the computation of an optimal symmetric orthogonalization of a set of vectors. We suggest two iterative techniques to compute an inverse square root of a given matrix. The two schemes are analyzed and their numerical stability properties are investigated.
TL;DR: In this article, the authors obtained a majorization inequality which relates the singular values of a complex square matrix A and those of T A where T A is a linear map, and they generalized several known majorization inequalities for eigenvalues and singular values.
TL;DR: In this paper, it was shown that a square matrix M has the property that M ϰ = 0 whenever ϰ T Mϰ =0 if and only if it can be decomposed into the form E T AE, for some matrix E and some matrix A that is either positive definite or negative definite.
TL;DR: For a certain class of n-dimensional Toeplitz-plus-Hankel systems of equations, an efficient method of solution is presented and an example is given to illustrate the similarity transforms.
Abstract: Two unitary matrices are presented that transform a Hermitian Toeplitz matrix into a real Toeplitz-plus-Hankel matrix and vice versa. Additional properties and consequences of these unitary transformations are also presented. For a certain class of n-dimensional Toeplitz-plus-Hankel systems of equations, an efficient method of solution is presented. An example is given to illustrate the similarity transforms. Toeplitz-plus-Hankel systems arise in a wide variety of applications, such as linear filtering theory, discrete inverse scattering, and discretization of certain integral equations arising in mathematical physics. >
01 Jan 1991
TL;DR: In this paper, the eigenvalues of the rate matrix are determined through a Localization Theorem, and the eigvectors and generalized left eigenvectors are then determined to arrive at the Jordan canonical form representation.
Abstract: Nonlinear matrix equations of the form where Fi, i=0, 1,2 ... ,m are known nxn nonnegative sub-matrices of a state transition matrix arise ubiquitously in the analysis of various stochastic models utilized in queueing, inventory and communication theories. Computation of the minimal nonnegative solution of (1), called the rate matrix R′, is essential for the equilibrium analysis of these models. Previously, this matrix has been computed by iterative techniques. In this paper, an analytical method for the computation of the rate matrix is proposed. Specifically, the eigenvalues of the rate matrix are determined through a Localization Theorem. The eigenvectors and generalized left eigenvectors are then determined to arrive at the Jordan canonical form representation of the rate matrix. Also, the problem of computation of the rate matrix is formulated as a linear programming problem. This method can be adapted in a natural manner for the equilibrium analysis of M/G/l type Markov chains.
Proceedings Article•
01 Jan 1991
Patent•
28 Jun 1991TL;DR: In this article, a matrix multiplier circuit based on distributed arithmetic is proposed, where the maximum advantage of parallel processing and pipelined processing is achieved if N is the nearest integer to M/J where M is the precision of the elements from the data matrix.
Abstract: A matrix multiplier circuit which is based on distributed arithmetic is disclosed. In the conventional matrix multiplier circuit, one row of J elements from a data matrix is multiplied in parallel with K columns of elements from a transform matrix to form one row of elements of an output matrix. In contrast, in the inventive matrix multiplier circuit, N rows of elements from the data matrix are multiplied in parallel with K columns of elements from a transform matrix. Maximum advantage of parallel processing and pipelined processing is achieved if N is the nearest integer to M/J where M is the precision of the elements from the data matrix.
TL;DR: In this paper, it was shown that the quadratic form in an elliptically contoured matrix variate has a constant rank and its nonzero eigenvalues are distinct with probability one if the matrix distribution satisfies certain conditions.
TL;DR: This work investigates the communication complexity of singularity testing, where the problem is to determine whether a given square matrix M is singular, and shows that, for n × n matrices of k-bit integers, the communication complex of Singularity Testing is Θ(k n2).
TL;DR: In this paper, the Lyapunov method for determining the inertia of a matrix in terms of inertia of solutions of a certain linear matrix equation is extended to matrix polynomials.
Abstract: The Lyapunov method for determining the inertia of a matrix in terms of inertia of solutions of a certain linear matrix equation is extended to matrix polynomials.Generalization of well-known inertia theorems are obtained using the recently developed concept of Bezoutian for several matrix polynomials.
11 Dec 1991
TL;DR: In this article, the robust stability in a real parameter space for robot polynomial type spaces and matrix type spaces was studied. But the robustness was not considered in this paper.
Abstract: The authors report on a study of robust stability in a real parameter space for robot polynomial type spaces and matrix type spaces. Using the Sylvester resultant matrix or the Kronecker sum the robust stability question can be converted into a generalized eigenvalue problem of a matrix pencil. Some sufficient and necessary conditions are given. The admissible perturbation set is also defined. This set can be found via a generalized eigenvalue computation. A method is proposed to compute a polytope to approximate a maximal admissible perturbation set via a matrix measure. Some results can be extended to the discrete-time case. >
TL;DR: In this paper, the quantum R matrix for E7 and F4 is explicitly constructed for defining representations 56 and 26 of E 7 and F 4, respectively, for these, Skein relations, link polynomials, and spectral parameter dependent solutions of Yang-Baxter equation are obtained.
Abstract: The quantum R matrix for E7 and F4 are explicitly constructed for the defining representations 56 and 26 of E7 and F4, respectively. For these, Skein relations, link polynomials, and spectral parameter‐dependent solutions of Yang–Baxter equation are obtained. The R matrix for the matrix elements related by the Weyl reflection has different values surprisingly.
TL;DR: In this article, it was shown that the condition number of AD 1 can be at least (n/2) 1/2 times the minimal one, where n is the number of columns in the matrix.
Abstract: Optimal diagonal scaling of an n×n matrix A consists in finding a diagonal matrix D that minimizes a condition number of AD. Often a nearly optimal scaling of A is achieved by taking a diagonal matrix D 1 such that all diagonal elements of D 1 AT AD 1 are equal to one. It is shown in this paper that the condition number of AD 1 can be at least (n/2)1/2 times the minimal one. Some questions for a further research are posed.