Showing papers on "Matrix decomposition published in 1975"

Matrix Factorization over $GF(2)$ and Trace-Orthogonal Bases of $GF(2^n )$

Abstract: The main result of this paper is a theorem showing that every binary, symmetric matrix A can be factored over $GF(2)$ into $A = BB'$, where the number of columns of B is bounded from below by either the rank $\rho (A)$ of A, or by $\rho (A) + 1$, depending on whether at least one, or none, of the main-diagonal entries of A is nonzero. An algorithm for a minimal factorization of a given matrix A and an application of this result for finding a trace-orthogonal basis of $GF(2^n )$ are presented.

On the factorization of a nonsingular rational matrix

Abstract: Given a multiport transfer function A , the question arises whether it can be realized using feedback loops of shortest possible length. The most elegant way to achieve this is to realize A in cascade form, with each subsystem in the cascade of smallest possible degree. Any transfer matrix of finite degree is a matrix whose entries are rational functions of the complex variable p , and a cascade synthesis of such a p -matrix will bear heavily on the properties of the factorization of rational p -matrices into simpler factors. We will call a p -matrix of the first degree an elementary factor. It is shown that 1) a nonsingular rational p -matrix can be factored into a minimal product of elementary factors provided either all poles or zeros are of the first order but not necessarily of the first degree; 2) that any nonsingular matrix can be factored into a product of elementary factors with the bound on the number of factors being a function of the degree of the matrix; and 3) that there are matrices which cannot be factored out minimally. Factorization procedures in special cases were first deduced by Belevitch and Youla, and later, using an improved criterion for degree reduction, by the first author. These procedures all use unitary or so-called J -unitary elementary factors. The theory presented here uses a general type of elementary factor to attack the problem of cascade decomposition of a general nonsingular transfer function. Thus the question whether a general (rational) multiport can be synthesized by means of feedback loops of length one has to be answered negatively although a very large number of systems can, and a criterion is given to distinguish the two cases. Also, an algorithm and an example are presented to exhibit how the factorization (if it exists) can be constructed.

Multirate sampled-data systems analysis via vector operators

Abstract: The primary difficulties of both the time-domain switch decomposition method and the frequency-domain decomposition method are overcome by introducing certain matrix operators and performing spectral factorization of resulting matrices of polynomials in the z -transform variable. Topological operations of the switch-decomposition method are simplified. This new approach eliminates the need to solve a system of equations with rational polynomial coefficients such as arises in the frequency-decomposition method. The determination of a multirate sampled-data system's characteristic polynomial no longer requires the evaluation of a determinant of rational polynomial elements. New results on obtaining modified z -transforms from standard z -transforms at a faster rate and vice versa are presented.

Matrix processors using p-ADIC arithmetic for exact linear computations

Abstract: A unique code (called Hensel's code) is derived for a rational number, by truncating its infinite padic expansion The four basic arithmetic algorithms for these codes are described and their application to rational matrix computations is demonstrated by solving a system of linear equations exactly, using the Gaussian elimination procedure A comparative study of the computational complexity involved in this arithmetic and the multiple prime module arithmetic is made with reference to matrix computations On this basis, a multiple padic scheme is suggested for the design of a highly parallel matrix processor

Extended system matrices-Transfer functions and system equivalence

Abstract: We introduce an extension of the usual notion of an input/output map and for linear constant-parameter systems an extended transfer function, that has a (matrix) polynomial- or more generally a differential/difference operator inverse, containing as a submatrix Rosenbrock's system matrix. A new class of system equivalence - "maximally strict system equivalence" (m.s.s.e.) is introduced as a necessary and sufficient condition for two minimal systems to have the same transfer function. We give an outline of the extension of these results to non-minimal systems, where results are only known for state-space systems in this generality. In the conclusion we discuss the relation of the extended system matrix to other deterministic and stochastic problems.

Sparse-matrix algorithm for transient analysis of nonlinear electrical networks

Abstract: A sparse-matrix algorithm appropriate to nodal admittance formulation of transient analysis of electrical network is described. The algorithm depends for its efficiency on the splitting of the coefficient matrix into two partitions. These partitions are decomposed sequentially as the LU factorisation of the matrix proceeds. In the decomposition of the first partition, adequate roundoff-error control is shown to be maintained, where pivots are selected from elements in the leading diagonal. In the second part, the algorithm selects suitable pivot elements by means of row reordering as the decomposition of the matrix is carried out. To reduce infill in coefficient matrix, the nodes associated with the first partition are renumbered once-for-all prior to the decomposition of the matrix. The algorithm is particularly suited to small machines having restricted word length and in which floating-point operations are performed by software routines. A comparison is made between the efficiencies of the present algorithm and a full-matrix decomposition method by means of transient analysis of a medium-size circuit.

A note on the LU factorization of a symmetric matrix

Abstract: So f rom (6) we have a linear system of n equat ions with n unknowns which generally possesses a unique Copyright @ 1975, Association for Computing Machinery, Inc. General permission to republish, but not for profit, all or part of this material is granted provided that ACM's copyright notice is given and that reference is made to the publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the Association for Computing Machinery. Author's address. University of Technology, Department of Mathematics, Loughborough, Leicestershire LE11 3TU, England.

On the decomposition of state space

Abstract: The response of a linear control system is often viewed as a superposition of independent, modal responses. For complex systems, traditional techniques for resolving modal responses may be either inapplicable, quite expensive, or numerically unstable. A new method, which incorporates the QR transformation with several properties of matrix equations, is proposed here as an alternative. The method effectively decomposes a system into a set of independent subsystems having the same natural frequencies as the original system.