Showing papers on "Matrix analysis published in 1976"
TL;DR: In this article, it was proved that a real symmetric tridiagonal matrix with positive codiagonal elements is uniquely determined by its eigenvalues and the eigen values of the largest leading principal submatrix, and can be constructed from these data.
149 citations
01 Aug 1976
TL;DR: In this article, generalized Bezoutian and Sylvester matrices are defined and discussed in a short paper, where the relationship between these two forms of matrices is established and it is shown that the degree of a real rational function can be ascertained by checking the rank of either one of these generalized matrices formed using a polynomial matrix fraction decomposition of the prescribed transfer function matrix.
Abstract: Generalized Bezoutian and Sylvester matrices are defined and discussed in this short paper. The relationship between these two forms of matrices is established. It is shown that the McMillan degree of a real rational function can be ascertained by checking the rank of either one of these generalized matrices formed using a polynomial matrix fraction decomposition of the prescribed transfer function matrix. Earlier established results by Rowe and Munro are obtained as a special case. Several theorems related to the rank testing and other properties of the generalized matrices are discussed and various research problems are listed in the conclusion.
119 citations
01 Jan 1976
TL;DR: The problem of interconnecting two multilevel subsystem models defined by binary matrices A and B and a common, transitive, contextual relation to form a system model defined by matrix M is solved and a method for finding this structure is given.
Abstract: The problem of interconnecting two multilevel subsystem models defined by binary matrices A and B and a common, transitive, contextual relation to form a system model defined by matrix M is solved. The entries of the unknown interconnection matrices X and Y are shown to form a multilevel implication structure. A method for finding this structure is given. The implication matrix that defines the structure furnishes a simple means of determining the inference opportunity of any unknown in X or Y at any point in the development of these matrices. Transitive bordering of A corresponds to the special case B = 1. When the system has many elements, it may be advisable to form a matrix A for a subset and then use transitive bordering iteratively to complete the structuring process.
60 citations
01 Dec 1976
TL;DR: In this article, a generalized resultant matrix and a fast algorithm for testing the coprimeness of two polynomial matrices, extracting their great common divisor, finding the McMillan degree and the observability indices of the associated minimal realization are presented.
Abstract: We present a generalized resultant matrix and a fast algorithm for testing the coprimeness of two polynomial matrices, extracting their great common divisor, finding the McMillan degree and the observability indices of the associated minimal realization.
57 citations
TL;DR: Tests on normal equation matrices encountered in adjustments of geodetic networks by least squares demonstrate that the algorithm produces significantly lower profiles than the widely used reverse Cuthill-McKee algorithm.
Abstract: An algorithm for improving the profile of a sparse symmetric matrix is introduced. Tests on normal equation matrices encountered in adjustments of geodetic networks by least squares demonstrate that the algorithm produces significantly lower profiles than the widely used reverse Cuthill-McKee algorithm.
52 citations
34 citations
TL;DR: In this article, a method to find the components of a Toeplitz matrix in close form when the matrix has only 2m + 1$ nonvanishing (central) diagonals is presented.
Abstract: A matrix is said to be of Toeplitz type if it has equal elements along diagonals. These matrices, with the additional property of symmetry, arise frequently in statistical work, as covariance matrices of wide-sense stationary stochastic processes, in nonparametric theory, etc. The inverse is often of interest, and a method, is developed to find its components in close form when the $T \times T$ matrix has only $2m + 1$ nonvanishing (central) diagonals $( {1\leqq m < T} )$. The method consists in posing difference equations for the components of the inverse, and solving them explicitly. The resulting procedures reproduce known results when $m = 1$, provide an expression for the inverse when $m = 2$, provide approximations in these two important cases and an interpretation for another approximation for general m, and are shown to be particularly suited for two methods of estimation in moving average models of time series. The inverse of a related matrix is also studied.
30 citations
TL;DR: The purpose of this correspondence is to give mnemotechnic rules for calculating gradient matrices, i.e., the first derivative of a scalar-valued function with respect to a matrix.
Abstract: The purpose of this correspondence is to give mnemotechnic rules for calculating gradient matrices, i.e., the first derivative of a scalar-valued function with respect to a matrix.
24 citations
01 Dec 1976
TL;DR: In this paper, new criteria as a generalization of the resultant of two polynomials are given for relative primeness of polynomial matrices together with system theoretic interpretations and unification of the existing and new criteria.
Abstract: New criteria as a generalization of the resultant of two polynomials are given for relative primeness of polynomial matrices together with system theoretic interpretations and unification of the existing and new criteria.
20 citations
01 Jan 1976-Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences
TL;DR: In this paper, the authors give a co mpl e te desc ripti on of the 3 X 3, re a l, D-s ta ble ma tri ces and th e re by exte nd one of th e main theore ms in 12].
Abstract: W e give a co mpl e te desc ripti on of the 3 X 3 , re a l, D-s ta ble ma tri ces and th e re by exte nd one of th e main theore ms in 12].1 One nice feature of o ur c ha rac te ri zin g co nditi ons is that the re qui s ite in var ian ce unde r ac ti o n by the multipl ic at ive group of diago nal matri ces with positive diago nal e ntri es a nd th e requi s ite in var ia nce unde r inve rs io n are c lea r. Our result is more e vidence that the se t of D-s ta ble matri ces is co mp li c a te d , a nd we ho pe th a t it will assis t th e sea rc h fo r a ge ne ral desc ripti on , b y he lpin g with th e formul a ti on of conjec ture s and by providin g counte rexa mples. For more info rm a ti o n on thi s exte ns ive ly s tudi e d prob le m see [3]. We ca ll th e square matrix M (pos itive) stable p rov id ed that Re( .\\ ) > 0 fo r e ve ry eige nva lu e .\\ of M_ And M is D-s tab le provide d th a t MD is s tab le for e ve ry di ago na l ma trix /J whose diago nal e nt ri es are pos itive . Le t M = (m ij ) be a 3 X 3 matrix with re a l e ntri es . Th e princ ipal minors of M will be de noted : a = ml l , 6 = 111 11, c= m :t:t, A = m~~ m:l:!m :! t ln 1:J' B = m\"In:!3In :! ,lnt 3 , C = I11llln22m 2 tlnI 2, o = de t (M). W e s ha ll say th a t a a nd A , 6 a nd 8 , c a nd C, and 1 a nd 0 are s upple me nta ry princ ipal minors of each oth e r. W e s ay that M is of type 1 if so me pr in c ipa l minor of M vani shes without its s upple me nt vani s hing a lso. O th e rwise M is of t ype 2_ Le t CI = v-;A + VbB + V\"£_ THEOREM: M is D-s ta6le if and only if
17 citations
TL;DR: In this paper, the Hankel matrix associated with a polynomial f(x) can be used to obtain information on the location of the zeros of f (x) in a given half plane.
Abstract: It is shown how the well-known Hankel matrix associated with a polynomial f(x) can be employed to obtain information on the location of the zeros of f(x) in a given half plane and in particular to obtain a test for aperiodicity of f(x) .
TL;DR: In this paper, the authors studied inclusion relations between four classes of matrices: normal matrices, matrices with equal spectral radius and spectral norm, and matrices whose numerical range coincides with the convex polygon spanned by their eigenvalues.
Abstract: In this paper we study inclusion relations between the following four classes of matrices: normal matrices, matrices with equal spectral radius and spectral norm, matrices whose numerical range coincides with the convex polygon spanned by their eigenvalues, and matrices with equal numerical and spectral radii.
TL;DR: In this article, the forms of the following linear transformations are determined: the linear transformations that hold either the trace or the second elementary symmetric function of the eigenvalues of each matrix fixed, and in addition preserve either the determinant, or the permanent, or an elementary symmetry of the squares of the singular values.
Abstract: The general problem considered is: what linear transformations on matrices preserve certain prescribed invariants or other properties of the matrices? Specifically, the forms of the following linear transformations are determined: the linear transformations that hold either the trace or the second elementary symmetric function of the eigenvalues of each matrix fixed, and in addition preserve either the determinant, or the permanent, or an elementary symmetric function of the squares of the singular values, or the property of being a rank 1 matrix or a unitary matrix.
TL;DR: This method suggests a quick analysis of networks avoiding any matrix analysis or matrix inversion techniques, as in usual practice, by direct evaluation of the determinant and the appropriate cofactors of the node-admittance matrix of the network.
Abstract: A method is proposed for computing the network transfer functions by direct evaluation of the determinant and the appropriate cofactors of the node-admittance matrix of the network using rules governed by topological formulae but without actually generating trees of the corresponding graph. This method suggests a quick analysis (and eventually design) of networks avoiding any matrix analysis or matrix inversion techniques, as in usual practice.
TL;DR: In this article, a matrix vector formalism is developed for systematizing the manipulation of sets of nonlinear algebraic equations, all manipulations are performed by multiplication with specially constructed transformation matrices.
01 Jan 1976
TL;DR: In this article, the authors discuss vector and matrix operations for multivariate analysis, and the concept of the determinant of a matrix, which is a special case of scalar multiplication.
Abstract: Publisher Summary This chapter discusses vector and matrix operations for multivariate analysis. Facility in the arithmetic of vectors and matrices is essential in multivariate analysis. The chapter reviews the fundamentals of vector and matrix operations, and the concept of the determinant of a matrix. Multivariate analysis makes liberal use of vector concepts from linear algebra. Two vectors of the same order are equal if they are equal component by component. Scalar multiplication of a vector involves multiplying each component of the vector by the scalar. The chapter describes vectors as ordered n -tuples of numbers that are subject to certain manipulative rules. Two matrices A and B are equal only if they are of the same order and each entry of the first is equal to the corresponding entry of the second. In matrix addition, each entry of a sum matrix is the sum of the corresponding entries of the two matrices being added. Matrices can also be multiplied by a number and this is called—scalar multiplication of the matrix.
TL;DR: In this paper, it was shown that the synthesis of closed-loop linear multivariable discrete-time systems can be directly effected by performing equivalence transformations on appropriate polynomial matrices.
Abstract: It is shown that the synthesis of closed-loop linear multivariable discrete-time systems can be directly effected by performing equivalence transformations on appropriate polynomial matrices. These polynomial matrices are the Smith canonical forms of the closed-loop characteristic matrices of such systems subject to the constraints imposed by the fundamental theorem of linear state-variable feedback.
TL;DR: An algorithm is given for computing the inverse of a linear time-invariant multivariable system that reduces the problem to that of inverting a polynomial matrix, and hence requires less computations.
Abstract: An algorithm is given for computing the inverse of a linear time-invariant multivariable system. The algorithm, which is an extension of an existing algorithm, reduces the problem to that of inverting a polynomial matrix, and hence requires less computations. A more efficient test for determining whether a system is invertible is also given.
TL;DR: In this article, it is shown that chain and chain scattering matrices of linear, time-invariant, passive networks are J-contractive real matrices and necessary and sufficient conditions are given for their state-space representation.
Abstract: Necessary and sufficient conditions are given for J-contractive real matrices in terms of their state-space representation. It is shown that chain and chain scattering matrices of linear, time-invariant, passive networks are J-contractive real matrices.