Showing papers on "Matrix analysis published in 1975"
21 citations
TL;DR: In this article, the necessary and sufficient conditions for the characteristic roots of a matrix A to lie inside the unit circle are presented, and the following conditions are proved: 1) Linear combinations b i of the coefficients of coefficients of λ √ n of A.
Abstract: In this short paper several theorems related to the necessary and sufficient conditions for the characteristic roots (eigenvalues) of a matrix A to lie inside the unit circle are presented In particular, the following conditions are proved 1) Linear combinations b i of the coefficients of λiin
21 citations
TL;DR: In this paper, it was shown that the test to determine whether all eigenvalues of a complex matrix of order n lie in a certain sector can be replaced by an equivalent test to find whether all the eigen values of a real matrix with order 4n lie in the left haft plane.
Abstract: It is shown that the test to determine whether all eigenvalues of a complex matrix of order n lie in a certain sector can be replaced by an equivalent test to find whether all eigenvalues of a real matrix of order 4n lie in the left haft plane.
18 citations
TL;DR: The results of this paper comprise a study of a special class of combinatorial matrices called chainable matrices, where the structure and algebraic behavior of these matrices are given specific attention.
13 citations
TL;DR: In this paper, the identification of a linear time-invariant discrete system represented by a matrix transfer function in the form of a matrix fraction is discussed, and all matrix transfer functions satisfying given input-output observations are determined.
Abstract: This short paper concerns the identification of a linear time-invariant discrete system represented by a matrix transfer function in the form of a matrix fraction. The relations between external and internal descriptions are discussed, and all matrix transfer functions satisfying given input-output observations are determined.
12 citations
TL;DR: An algorithm is presented for transposing large nonsquare matrices stored externally or in core and a fast algorithm of Eklundh and direct algorithm are applied to partitioned matrices and combined to effect the transpose of a nonsquared matrix.
Abstract: An algorithm is presented for transposing large nonsquare matrices stored externally or in core. A fast algorithm of Eklundh and direct algorithm are applied to partitioned matrices and combined to effect the transpose of a nonsquare matrix. The matrix must be augmented so that its dimensions have a large common divisor. In the case of large externally stored matrices, additional external storage for a matrix of equal size is not required.
11 citations
TL;DR: In this paper, a matrix vector formalism is developed for systematizing the manipulation of sets of non-linear algebraic equations, all manipulations are performed by multiplication with specially constructed transformation matrices.
9 citations
TL;DR: The algorithm neither requires cofactor calculation, nor applies any state transformation, and the only requirement is calculation of the coefficients of characteristic equations of given matrices.
Abstract: Given the state space realization \{A,B,C\} of a linear time-invariant multivariable dynamic system, a computational algorithm for calculating elements of the corresponding transfer function matrix is presented. The algorithm neither requires cofactor calculation, nor applies any state transformation. The only requirement is calculation of the coefficients of characteristic equations of given matrices. Also, the Fortran subroutine based on the presented algorithm is included.
7 citations
TL;DR: In this paper, a necessary condition for a linear combination of permutation matrices to be an orthogonal matrix is that the sum of the coefficients in the linear combination be ± 1.
7 citations
Journal Article•
7 citations
01 May 1975
TL;DR: In this paper, an analytical function of a companion matrix with multiple eigenvalues is expressed as the linear combination of constituent matrices and it is found that the constituent matrix is simply the product of the two matrices of the Vandermonde matrix.
Abstract: An analytical function of a companion matrix with multiple eigenvalues is expressed as the linear combination of constituent matrices It is found that the constituent matrix is simply the product of the two matrices whine elements are part of the entries of Vandermonde matrix and its inverse Calculation of these values is given as simple, explicit, and recunent formulas, and is very useful for both hand and machine computations
01 Mar 1975
TL;DR: In this paper, the stability of polynomial matrices is tested by mapping positive real parts of the matrix to the exterior of the unit circle and applying the power method to the resulting matrix.
Abstract: A scheme is proposed for testing the stability of polynomial matrices. Eigenvalues with positive real parts are mapped onto the exterior of the unit circle, and instability is subsequently traced by application of the power method to the resulting matrix. The difficulties of singularity of the coefficient matrices involved are overcome, since inversions are avoided, and any sparsity can be efficiently handled.
TL;DR: In this article, it was shown how the well-known Schwarz's result about the location of eigenvalues of a matrix in left-half plane can be derived by using an inertia theorem due to Chen.
Abstract: It is shown how the well-known Schwarz's result about the location of eigenvalues of a matrix in left-half plane can be derived by using an inertia theorem due to Chen.
TL;DR: The feasibility of factorizing non-negative definite matrices with elements that are rational functions of several real variables and having real coefficients is discussed in this article, where polynomial matrices are considered briefly.
01 Sep 1975
TL;DR: In this article, the Lagrange-Sylvester (LS) theorem is used to simplify the reduction of matrix functions without explicit use of the LS theorem and of the characteristic roots.
Abstract: Straightforward reduction of matrix functions without explicit use of the Lagrange-Sylvester (LS) theorem and of the characteristic roots is achieved by expansions in terms of Lucas polynomials. Use of the LS theorem within the framework of the Lucas polynomial method and, furthermore, of circulant matrices leads to substantial simplifications.
TL;DR: In this paper, extreme value results for concave and convex symmetric functions of the eigenvalues of B + P ∗ AP as functions of partial isometry P were obtained.
TL;DR: In this article, the authors considered the determination of the feedback coefficients for a linear time-invariant, system, having one control input, such that, the characteristic roots of the closed-loop system matrix lie at specified points in the complex plane.
Abstract: This work considers the determination of the feedback coefficients for a linear time-invariant, system, having one control input, such that, the characteristic roots of the closed-loop system matrix lie at specified points in the complex plane. This is achieved by considering the relationships between the eigenvalues and the traces (of various orders) of a matrix. The process merely requires the solution of linear algebraic equations and is easily computerized.
01 Jan 1975
TL;DR: A simple method for the generation of the constituent matrices when multiple eigenvalues are present is outlined and it is shown that an appropriate resolving polynomial for themultiple eigenvalue is needed.
Abstract: A simple method for the generation of the constituent matrices when multiple eigenvalues are present is outlined. The method is based on the generation of an appropriate resolving polynomial for the multiple eigenvalue.
TL;DR: In this article, an iterative method for the solution of the generalized eigenproblem is proposed, which is shown to possess cubic order of convergence in the case of distinct eigenvalues.
Abstract: An iterative method is suggested for the solution of the generalized eigenproblem. The iteration is shown to possess cubic order of convergence in the case of distinct eigenvalues. The algorithm appears at its best advantage when good starting values are available; engineering applications where this is so are briefly discussed.
TL;DR: Some computational aspects of matrix generalized inversion for computer manipulations which provide partial relief to storage difficulties, along with a discussion of some applications, particularly in the area of statistical analyses.
Abstract: Numerous theoretical problems have been simplified and, in many cases, unified via the theory of matrix generalized inversion, but not withstanding storage difficulties, occasionally, at the time of computer utilization. The sizes of matrices encountered may, in some cases, be too large for standard manipulations using existing routines. The purpose of this paper is to present some computational aspects of matrix generalized inversion for computer manipulations which provide partial relief to such problems, along with a discussion of some applications, particularly in the area of statistical analyses. Partial relief may be had, for example, through the utilization of the matrix kronecker product and its associated properties, special storage processes such as those available in manipulations with sparse matrices, and schemes for computing the matrix generalized inverse of partitioned matrices. In particular, a procedure for recursive partitioning is developed which permits the Moore Penrose inversion of matrices of any (finite) order.
16 May 1975
TL;DR: The program given in this report calculates the state transition matrix as a linear combination of real time functions starting from any real system matrix.
Abstract: : Many programs are available which calculate numerically the state transition matrix at a specific time. However, programs which give the solution as a linear combination of real time functions are not generally available. The program given in this report calculates the state transition matrix as a linear combination of real time functions starting from any real system matrix.
TL;DR: In this paper, it was shown that an n-dimensional vector whose components are complex numbers can be visualized as a plane ellipse in geometric n-space, and that the length and direction of a complex vector are interpreted in terms of the ellipsse axes thus giving a deeper insight into the eigenstructure of system transfer matrices.
Abstract: It is shown that an n -dimensional vector whose components are complex numbers can be visualized as a plane ellipse in geometric n -space. The length and direction of a complex vector are interpreted in terms of the ellipse axes thus giving a deeper insight into the eigenstructure of system transfer matrices. The system characteristic ellipse is introduced and its application to the measurement of characteristic loci is discussed.
TL;DR: In this paper, a simple stability condition is obtained for a class of matrices which occur in the application of linear differential inequalities, where the matrices are linear matrices and linear inequalities are linear inequalities.
Abstract: A simple stability condition is obtained for a class of matrices which occur in the application of linear differential inequalities.