Showing papers on "Square matrix published in 1972"

On complex eigenvalues ofM andP matrices

Abstract: Inequalities are obtained for the complex eigenvalues of anM matrix or aP matrix which depend only on the order of the matrix.

The generalized inverse of a nonnegative matrix

Abstract: Necessary and sufficient conditions are given in order that a nonnegative matrix have a nonnegative MoorePenrose generalized inverse.

Information matrix for a mixture of two normal distributions

Abstract: This paper presents a numerical method for computation of the Fisher information matrix about the five parameters of a mixture of two normal distributions. It is shown, by using a simple transformation which reduces the number of parameters from five to three, that the computation of the whole information matrix leads to the numerical evaluation of a particular integral. The Hermite-Gauss quadrature formula, Romberg's algorithm, a power series, and Taylor's expansion are applied for the evaluation of this integral and the results are compared with each other. A short table has been provided from which the approximate information matrix can be obtained in practice.

Matrix print rotation

Abstract: A printer using a matrix of printing elements arranged in a square configuration with the printing elements being used to print alpha-numeric data in either a vertical or horizontal orientation by electronically selecting a rectangular matrix from less than the full number of printing elements in the square matrix to permit selective orientation of the printed data without mechanically reorienting the print head. The print element drive circuitry also enables the printing matrix to print from either end of the selected rectangular print matrix in either the horizontal or vertical orientation to provide four possible orientations of the printed alpha-numeric data. A memory is used to store the input control signals for each of the rows of the rectangular printing matrix while a control means is provided for reading out the control signals in either direction from the memory in combination with a matrix selection control to provide energization of a rectangular print matrix in either the horizontal or vertical configuration.

A fundamental matrix equation for finite sets

Abstract: Let S={x1, x2, * * *, x,} be an n-set and let Sl, S2,. .. , Sm be subsets of S. Let A of size m by n be the incidence matrix for these subsets of S. We now regard x1, x2, .. , xn as independent indeterminates and define X=diag[x1, x2, *.. *, xnj. We then form the matrix product AXAT= Y, where AT denotes the transpose of the matrix A. The symmetric matrix Y has in its (i,j) position the sum of the indeterminates in S,r-Sj and consequently Ygives us a complete description of the intersection patterns S,(Si. The specialization x1=x2= ... =xn=1 of this basic matrix equation has been used extensively in the study of block designs. We give some other interesting applications of the matrix equation that involve subsets with various restricted intersection patterns. 1. The matrix equation. Let S= {x1, x2, * * , xj} be an n-set (a set of n elements) and let S,, S2,.. * , Sm be subsets of S. We set aij =1 if xi is a member of Si and we set aij=O if xj is not a member of Si. The resulting (0, 1)-matrix (1.1) A = [aij] of size m by n is the familiar incidence matrix for the subsets S,, S2, * , Sm of S. It is clear that A characterizes the configuration of subsets. Now let us regard xl, x2, * * *, xn, as independent indeterminates over the field of rational numbers and define (1.2) X= diag[xl, x2, ,xJ]. We then form the matrix product (1.3) AXAT Y The matrix AT denotes the transpose of the matrix A. The matrix Y is a symmetric matrix of order m. We know the structure of this matrix Received by the editors August 25, 1971. AMS 1970 subject classi{/cations. Primary 05B20, 05B30; Secondary 15A24.

Canonical Form for Matrices Under Unitary Congruence Transformations. I: Conjugate-normal Matrices

Abstract: A unitary congruence transform of a given square matrix A is $UA\tilde U$, where U is a unitary matrix, and $\tilde U$ is its transpose. The problem of finding the canonical (i.e., the simplest) matrix $A^0 $ in the set of all unitary congruence transforms of A is solved for matrices which satisfy $AA^ + = ( {A^ + A} )^ * $, $A^ + $ and $A^ * $ being the adjoint and the complex conjugate of A. Such matrices are called conjugate-normal. Symmetric, skew-symmetric and unitary matrices are special cases, but do not exhaust all conjugate-normal ones. The canonical form $A^ 0 $ turns out to be the direct sum of a diagonal matrix and of several $2 \times 2$ matrices of the form $( {\begin{array}{*{20}c} 0 & {| a |} \\ a & 0 \\ \end{array} } )$ . A particular matrix Z and the general form $Z'$ of U which transforms A into $A^0 $ by congruence are derived.

Recursive evaluation of padé approximants for matrix sequences

Abstract: An algorithm is described for calculating the existing Pade approximants to any sequence A0, A1,... of s × t-matrices. As an application the algorithm gives a new way for finding the minimal polynomial of any square matrix A and the inverse of the characteristic matrix xl - A.

Nonnegative matrices with nonnegative inverses

Abstract: We generalize a result stating that a nonnegative finite square matrix has a nonnegative inverse if and only if it is the product of a permutation matrix by a diagonal matrix. We consider column-finite infinite matrices and give a simple proof using elementary ideas from the theory of partially ordered linear algebras. In [1] the authors show that a nonnegative square matrix has a nonnegative inverse if and only if its entries are all zero except for a single positive entry in each row and column. In this note we generalize this result and simplify the proof as well. Let A denote the real linear algebra of all column-finite infinite matrices with real entries. We partially order A as follows: [aij] 10 and note that 1 +d_ (1 +d)2=yxyx

Inversion of certain symmetric band matrices

Abstract: A procedure is given for inversion of a symmetric band matrix with all elements in a certain diagonal equal. Starting from the main diagonal we have the elementsk,k−1,k−2, ... 2,1 with zeros in the remaining diagonals. Letting the second order difference operator δ2 operate on the rows of the matrix we obtain a new matrix which can be inverted by a partition method after certain permutations of the elements.

Matrix inequalities and the additive inverse eigenvalue problem

Abstract: LetA be anHermitiann×n matrix ands1, ...,sn real numbers; under what conditions does there exist a diagonal realn×n matrixM such thatA+M has eigenvaluess1, ...,sn? In the present note we prove a matrix inequality which gives a necessary condition for this problem to have a solution.

Solution of certain matrix equations

Abstract: The main purpose of this paper is to obtain solutions of matrix equations of the following types, AX-XB=C, XDX+AX+XB+C=O, in which case X is an unknown n by n matrix and A, B, C, D are n by n matrices having elements belonging to the field C of complex numbers. Results obtained extend those of W. E. Roth, J. E. Potter and others concerning the existence and the representation of solutions X of the above equations.

A simple proof of the Fisher-Fuller theorem

Abstract: Several years ago, Fisher and Fuller(1) proved that if P is a real square matrix with all members of its ‘nested set’ of principal minors non-zero, then there exists a real diagonal matrix, D , such that the characteristic roots of DP are all real, negative and distinct. This interesting and powerful result, used by the authors to derive further results concerning convergence of linear iterative processes, has since also proved of interest to economists studying the stability of economic general equilibrium.

Weak Spectral Inverses

Abstract: The notion of a weak spectral inverse of a square matrix is introduced, generalizing the definition of a spectral inverse given by Greville. A representation for weak spectral inverses in terms of the Drazin inverse is developed, and it is shown that a formula given previously for the Cline inverse is a special case. Necessary and sufficient conditions are given for a weak spectral inverse to be a spectral inverse, first in terms of the Drazin inverse representation, and then by making use of the quasi-commuting inverse defined by Erdelyi.

A Note on Jacobson Chains

Abstract: In this note we present a construction for the Jacobson chains of columns from the given elementary divisors of a square matrix A over a field $\mathcal{F}$, that is not necessarily algebraically closed. It generalizes the algorithm used by Gantmacher for the closed field case and does not assume a knowledge of the second canonical form under similarity.

Triangular partitioning for matrix inversion

Abstract: A triangular partitioning (additive) scheme for inverting a non-singular matrix is suggested. This scheme is direct and expresses the inverse matrix as a product of a lower triangular matrix and a set of upper triangular matrices

On the inversion of a particular matrix

Abstract: Necessary and sufficient conditions for the nonsingularity of a particular mn \times mn partitioned matrix C are given in terms of lower order m \times m matrices. The matrix C is formed by adding a block diagonal matrix to a partitioned matrix, each of whose submatrices are identical. Analytical formulas are given for both the determinant and the inverse of C .

Properties of the generalised inverse matrix in the electrical-network problem

Abstract: A modified form of Roth's transformation diagram for a linear graph is used to illustrate the solution of the electrical-network problem. The diagram illustrates particularly the significance of the orthogonal projections of the branch space into the branch voltage and current subspaces, which are defined by Kirchhoff's laws, and also the existence of the constrained matrix inverse which forms a basis for the solution of the electrical-network problem. Properties of the generalised inverse matrix are also discussed in relation to the network problem.

A remark on Latin squares and block designs

Abstract: Throughout this paper any matrix is square and of order q unless otherwise specified. I and J will represent an identity matrix and a square matrix with every element + 1 respectively; if necessary, the size will be indicated by a subscript. The Kronecker product of matrices A = ( a ij ) and B is the block matrix whose ( i, j )-th block is a ij B , and is written A x B .

On the Dispersion Matrix of the Maximum Likelihood Estimators of the Parameters of a Dynamic Model.

Abstract: : The approximation of a Toeplitz matrix by a circulant matrix is used to derive the dispersion matrix of the parameters of a dynamic model when the error process is known and when one has an approximation of the value of the parameters. (Author)

On a transformation operator for a system of Sturm-Liouville equations

Abstract: We prove the existence of a transformation operator with a condition at infinity that sends a solution of the matrix equation−y″ + My=λ2y (M is a constant Hermitian matrix) into a solution of the matrix equation−y″+Q(x)y+My=λ2y (the matrix function Q(x) is continuously differentiable for 0 ≤ x<∞ and it is Hermitian for each x belonging to [0, ∞)); we study some properties of the kernel of the transformation operator.

On a characterisation of matrix functions which are differences of two monotone matrix functions

Abstract: The class of matrix functions of 'bounded variation' was introduced by 0. Dobsch in a paper published in 1937 [2]. The consideration of this class of functions immediately gives rise to the consideration of those matrix functions of order n on an interval [a, b] that are representable as the difference of two monotone matrix functions on that interval. Such a difference will have high regularity properties when n is large and is therefore much more than simply a function of bounded variation. The characterization of this class was sought in the paper of Dobsch [2]. The purpose of this paper is to give a complete description of a related class: the functions defined on (-1, 1) which have restrictions to any closed subinterval which are such differences.