Showing papers on "Triangular matrix published in 1995"
TL;DR: This survey paper describes how strands of work that are important in two different fields, matrix theory and complex function theory, have come together in some work on fast computational algorithms for matrices with what the authors call displacement structure, and develops a fast triangularization procedure.
Abstract: In this survey paper, we describe how strands of work that are important in two different fields, matrix theory and complex function theory, have come together in some work on fast computational algorithms for matrices with what we call displacement structure. In particular, a fast triangularization procedure can be developed for such matrices, generalizing in a striking way an algorithm presented by Schur (1917) [J. Reine Angew. Math., 147 (1917), pp. 205–232] in a paper on checking when a power series is bounded in the unit disc. This factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra. We mention, among others, inverse scattering, analytic and unconstrained rational interpolation theory, digital filter design, adaptive filtering, and state-space least-squares estimation.
447 citations
TL;DR: In this article, Bruckner showed that a sequence S = {sn} converges to L statistically (T) if and only if "most" of the subsequences of S converge, in the ordinary sense, to L. Corresponding results for lacunary statistical convergence are also presented.
Abstract: The concept of statistical convergence of a sequence was first introduced by H. Fast. Statistical convergence was generalized by R. C. Buck, and studied by other authors, using a regular nonnegative summability matrix A in place of C1 The main result in this paper is a theorem that gives meaning to the statement: S = {s,,} converges to L statistically (T) if and only if "most" of the subsequences of S converge, in the ordinary sense, to L. Here T is a regular, nonnegative and triangular matrix. Corresponding results for lacunary statistical convergence, recently defined and studied by J. A. Fridy and C. Orhan, are also presented. INTRODUCTION The concept of the statistical convergence of a sequence of reals S = {sn was first introduced by H. Fast [9]. The sequence S = {sn} is said to converge statistically to L and we write lim s, = L (stat) if for every e > , n-+oo rim n-'){k e1 = 0, where JAI denotes the cardinality of the set A. Properties of statistically convergent sequences were studied in [5, 6, 12, and 16]. In [13] Fridy and Miller gave a characterization of statistical convergence for bounded sequences using a family of matrix summability methods. Statistical convergence can be generalized by using a regular nonnegative summability matrix A in place of C,. This idea was first mentioned by R. C. Buck [3] in 1953 and has been further studied by Sember and Freedman ([10 and 11]) and Connor ([5 and 7]). Regular nonnegative summability matrices turn out to be too general for our purposes here, instead we use the concept of a mean. A matrix T = (amn) will be called a mean if amn > 0 when n m, E'iamn = 1 for all m and limm,0oam, =0 for each n . If T = (amn) is a mean, following Buck, a sequence S = {sn} is said to be statistically T-summable to L and we write Sn L (stat T) if for every e > 0 Received by the editors August 18, 1993 and, in revised form, February 14, 1994; originally communicated to the Proceedings of the AMS by Andrew Bruckner. 1991 Mathematics Subject Classification. Primary 40D25; Secondary 40G99, 28A12. ? 1995 American Mathematical Society 0002-9947/95 $1.00 + S.25 per page
266 citations
TL;DR: In this paper, the authors show that this splitting plays a prominent role in the construction of Toda symmetries and their action on τ−functions; it also plays a crucial role in obtaining the Virasoro constraints for matrix integrals and ties up elegantly with the theory of orthogonal polynomials.
Abstract: into the algebras of skew-symmetric As and lower triangular (including the diagonal) matrices Ab (Borel matrices). We show that this splitting plays a prominent role also in the construction of the Toda symmetries and their action on τ−functions; it also plays a crucial role in obtaining the Virasoro constraints for matrix integrals and it ties up elegantly with the theory of orthogonal polynomials .
126 citations
TL;DR: In this paper, a technique for deriving bounds on the relative change in the singular values of a real matrix (or the eigenvalues of real symmetric matrix) due to a perturbation, as well as the angles between the unperturbed and perturbed singular vectors (or eigenvectors) was presented.
Abstract: A technique is presented for deriving bounds on the relative change in the singular values of a real matrix (or the eigenvalues of a real symmetric matrix) due to a perturbation, as well as bounds on the angles between the unperturbed and perturbed singular vectors (or eigenvectors). The class of perturbations considered consists of all $\delta B$ for which $B + \delta B = D_L BD_R $ for some nonsingular matrices $D_L $ and $D_R $. This class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off-diagonal block in a block triangular matrix. Many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations. Also some new relative perturbation and deflation results for the singular values and vectors of biacyclic, triangular, and shifted triangular matrices are presented.
112 citations
Patent•
25 Nov 1995
TL;DR: In this article, the network topology data is obtained in the form of a terminal adjacent matrix, converted into a triangular matrix with additional fixed reference paths, with formation of the matrix products for each line of the triangular matrix after its initialisation.
Abstract: The initialisation and actualisation system has the network topology represented by a network graph using a sparse vector method and a sparse matrix method. The network topology data is obtained in the form of a terminal adjacent matrix, converted into a triangular matrix with additional fixed reference paths, with formation of the matrix products for each line of the triangular matrix after its initialisation. A path table is then provided, with the matrix actualised in dependence on variations in the network topology, before actualisation of the path table and identification of the network graph components.
39 citations
TL;DR: It has been known for at least thirty years that certain nilpotent groups cannot be Kahler groups as discussed by the authors, i.e., fundamental groups of compact Kahler manifolds.
Abstract: It has been known for at least thirty years that certain nilpotent groups cannot be Kahler groups, i.e., fundamental groups of compact Kahler manifolds. The best known examples are lattices in the three-dimensional real or complex Heisenberg groups. It is also known that lattices in certain other standard nilpotent Lie groups, e.g., the full group of upper triangular matrices and the free k-step nilpotent Lie groups, k > 1, are not Kahler. The Heisenberg case was known to J-P. Serre in the early 1960’s, and unified proofs of the above statements follow readily from Sullivan’s theory of minimal models [6],[15], [19], or from Chen’s theory of iterated integrals [4], [10], or from more recent developments such as [9].
30 citations
TL;DR: An explicit formula for the number of upper triangular matrices, overGF(q)$, whose square is the zero matrix is proved.
Abstract: We prove an explicit formula for the number of $n \times n$ upper triangular matrices, over $GF(q)$, whose square is the zero matrix. This formula was recently conjectured by Sasha Kirillov and Anna Melnikov.
27 citations
TL;DR: In this article, direct limits of full triangular matrix algebras with order-preserving embeddings are studied. And the problem of characterizing full triangular matrices with order preserving embedding is investigated.
27 citations
TL;DR: A forward error bound is identified that holds not only for all the methods described here, but for any triangular equation solver that does not rely on algebraic cancellation; among the implications of the bound is that any such method is extremely accurate for certain special types of triangular systems.
Abstract: Several parallel algorithms have been proposed for the solution of triangular systems. The stability of four of them is analysed here: a fan-in algorithm, a block elimination method, a method based on a factorized power series expansion of the matrix inverse, and a method based on a divide and conquer matrix inversion technique. New forward error and residual bounds are derived, including an improvement on the bounds of Sameh and Brent for the fan-in algorithm. A forward error bound is identified that holds not only for all the methods described here, but for any triangular equation solver that does not rely on algebraic cancellation; among the implications of the bound is that any such method is extremely accurate for certain special types of triangular systems.
24 citations
TL;DR: In this paper, the eigenvalues of a natural random walk on the group of upper triangular matrices over the field of an odd prime, with 1's on the diagonal, are given.
Abstract: This paper gives sharp bounds on the eigenvalues of a natural random walk on the group of upper triangular $n \times n$ matrices over the field of characteristic $p$, an odd prime, with 1's on the diagonal. In particular, this includes the finite Heisenberg groups as a special case. As a consequence we get bounds on the time required to achieve randomness for these walks. Some of the steps are done using the geometric bounds on the eigenvalues of Diaconis and Stroock. However, the crucial step is done using more subtle and idiosyncratic techniques. We bound the eigenvalues inductively over a sequence of subspaces.
22 citations
TL;DR: It is shown that the problem of finding a similarity transformation G such that G?1AG is block diagonal with the condition number of G being at most ?
TL;DR: In this article, a simple determinantal equation for the boundary of all positive upper triangular matrices in B + is given. But this is not the case for all positive matrices with positive entries on the main diagonal.
TL;DR: In this paper, the authors derived the best lower triangular equivariant estimators of the normal covariance matrix and the precision matrix Σ-1 simultaneously, where the estimators are of the form, where Tis a lower triangular matrix and Dis a diagonal matrix of constants.
Abstract: In this paper we derive the best lower triangular equivariant estimators of the normal covariance matrix Σ and the precision matrix Σ-1 simultaneously. Our estimators are of the form , where Tis a lower triangular matrix and Dis a diagonal matrix of constants. We derive improved estimators which dominate . The risk of the derived estimators for p= 2 and 3 is also given.
09 Sep 1995
TL;DR: A parallel solution to the large sparse systems of linear equations is presented, based on a parallel pivoting technique for LU decomposition on a shared memory MIMD multiprocessor.
Abstract: A parallel solution to the large sparse systems of linear equations is presented. The solution method is based on a parallel pivoting technique for LU decomposition on a shared memory MIMD multiprocessor. At each application of the algorithm to the matrix several pivots for reducing the matrix in parallel are generated. During parallel pivoting steps only symmetric permutations are possible. Unsymmetric permutation for numerical stability however is possible during single pivoting steps. We will report on switching between parallel and single pivoting steps to assure numerical stability. Once the matrix is decomposed, the parallel pivoting information is used to solve structurally identical matrices repeatedly. The algorithms, their implementation, and the performance of the solution methods on actual multiprocessors are presented. Based on the resulting triangular matrix structure, two algorithms for back substitution are presented and their performance is compared.
TL;DR: In this paper, the authors studied the asymptotic behavior of splines obtained by orthogonalizing a large finite set of B-splines, in particular identifying the limiting profile when the knots are equally spaced.
Abstract: Cholesky factorization of bi-infinite and semi-infinite matrices is studied and in particular the following is proved. If a bi-infinite matrixA has a Cholesky factorization whose lower triangular factorL and its lower triangular inverse decay exponentially away from the diagonal, then the semi-infinite truncation ofA has a lower triangular Cholesky factor whose elements approach those ofL exponentially. This result is then applied to studying the asymptotic behavior of splines obtained by orthogonalizing a large finite set of B-splines, in particular identifying the limiting profile when the knots are equally spaced.
21 Jun 1995
TL;DR: It is shown that after a preliminary reduction of the matrices A and E to, respectively, an upper Hessenberg and an upper triangular matrix, the problem of computing the characteristic polynomial is transformed to the solution of certain triangular systems of linear algebraic equations.
Abstract: Presents an algorithm for computing the characteristic polynomial of the pencil (A-sE). It is shown that after a preliminary reduction of the matrices A and E to, respectively, an upper Hessenberg and an upper triangular matrix, the problem of computing the characteristic polynomial is transformed to the solution of certain triangular systems of linear algebraic equations. The authors show that the computed characteristic polynomial corresponds exactly to perturbed matrices A+/spl Delta/A and E+/spl Delta/E and the authors derive bounds for /spl Delta/A and /spl Delta/E. The authors also suggest how to improve on this backward error via iterative refinement.
TL;DR: In this article, the eigenvalues of random walks on groups of Cayley graphs were analyzed using geometric methods, and sharp bounds on the second largest eigenvalue of random walk on nilpotent groups with low class number were given.
Abstract: In this paper we discuss and apply a novel method for bounding the eigenvalues of a random walk on a group $G$ (or equivalently on its Cayley graph). This method works by looking at the action of an Abelian normal subgroup $H$ of $G$ on $G$. We may then choose eigenvectors which fall into representations of $H$. One is then left with a large number (one for each representation of $H$) of easier problems to analyze. This analysis is carried out by new geometric methods. This method allows us to give bounds on the second largest eigenvalue of random walks on nilpotent groups with low class number. The method also lets us treat certain very easy solvable groups and to give better bounds for certain nice nilpotent groups with large class number. For example, we will give sharp bounds for two natural random walks on groups of upper triangular matrices.
21 May 1995
TL;DR: In this paper the UDU/sup T/ decomposition of the generalized inertia matrix of an n-link serial manipulator as presented in symbolic form, where U and D, respectively, are the upper triangular and diagonal matrices.
Abstract: In this paper the UDU/sup T/ decomposition of the generalized inertia matrix of an n-link serial manipulator as presented in symbolic form, where U and D, respectively, are the upper triangular and diagonal matrices. To render the decomposition, the elementary upper triangular matrices, associated to a modified Gaussian elimination, are introduced, whereas each element of the inertia matrix is written as an expression, instead of finding it as a number with the aid of an algorithm. The resulting UDU/sup T/ decomposition shows recursive relations among the elements of the associated matrices. Thus, algorithms of order 'n' can be developed not only for the inverse but also for the forward dynamics. As an illustration, a forward dynamics algorithm is presented here.
TL;DR: In this article, the authors proved several results for the permanent of a nonnegative matrix over the max algebra, including Alexandroff inequality, Bregman's inequality, Cauchy-Binet formula and a Bebianotype expansion.
21 Aug 1995
TL;DR: It is shown how an elementwise algorithm can be reorganized in terms of blocked factorizations and higher level BLAS operations, and several ways to annihilate elements are compared.
Abstract: An algorithm for reduction of a regular matrix pair (A, B) to block Hessenberg-triangular form is presented. This condensed form Q T (A,B)Z = (H,T), where H and T axe block upper Hessenberg and upper triangular, respectively, and Q and Z orthogonal, may serve as a first step in the solution of the generalized eigenvalue problem Ax = λBx. It is shown how an elementwise algorithm can be reorganized in terms of blocked factorizations and higher level BLAS operations. Several ways to annihilate elements are compared. Specifically, the use of Givens rotations, Householder transformations, and combinations of the two. Performance results of the different variants are presented and compared to the LAPACK implementation DGGHRD, which indeed is unblocked.
TL;DR: In this article, the Jacobson radical of an infinite lexicographic product of upper triangular matrix algebras is determined, and connections between linear orderings and triangular operators are given.
Abstract: Some new connections are given between linear orderings and triangular operator algebras. A lexicographic product is denned for triangular operator algebras, and the Jacobson radical of an infinite lexicographic product of upper triangular matrix algebras is determined.
01 Jun 1995
TL;DR: Angelos et al. as mentioned in this paper showed that the product of the norm of the triangular truncation map on the n x n complex matrices with the distance from the normone hermitian nx n matrices to the nilpotents converges to 1/2.
Abstract: We show that, as n -0oo , the product of the norm of the triangular truncation map on the n x n complex matrices with the distance from the normone hermitian n x n matrices to the nilpotents converges to 1/2. We also include an elementary proof of D. Herrero's characterization of the normal operators that are norm limits of nilpotents. Suppose n is a positive integer and let 'n, $n, An denote, respectively, the sets of all n x n complex matrices, strictly upper triangular n x n matrices, and nilpotent n x n matrices. There is a natural mapping Tn: A4 -+ $n, namely, Tn (T) replaces the entries on or below the main diagonal of T with zeroes. The map Tn is called triangular truncation on 4n . On an infinite-dimensional space, the triangular truncation mapping does not always yield the matrix of a bounded operator. This is related to the fact that the range of the mapping that sends a bounded harmonic function on the unit disk to its analytic part is not included in HOO . For example, if f(z) = log(l z), then u = 2i Im(f) is bounded in modulus by ir, but the analytic part of u, namely f, is not bounded. In terms of Toeplitz operators, Tu is an operator with norm 7r, but the upper triangular truncation of Tu is the formal matrix for Tf, which is not a bounded operator. The matrix for Tu is the matrix whose (i, i)-entry is 0 and (i, j)-entry is 1/(j i) for 1 < i $ j < oc. For each positive integer n, let Tu n be the n x n upper-left-hand corner of Tu, i.e., the (i, i)-entry of Tu,n is 0, and the (i, j)-entry of Tu,n is 1/(j i) for 1 < i $ j < n . It follows that IITu,nII < Xr foreach n, andthat II T(T ,n)IT n oc as n -oc. Hence I ITnIoc as n oc . Much work has been done in determining IITn . S. Kwapien and A. Pelczynski [KP, pp. 45-48] proved in 1970 that liTnil = O(log(n)), K. Davidson [D, p. 39] proved that 4 < liminfn 00 liTnil/log(n), and, in 1993, J. R. Angelos, C. Cowen, and S. K. Narayan [ACN] proved that lim l-Tnll/log(n) = 1/7r. n--*oo Received by the editors September 22, 1993; this paper was presented at a miniconference in honor of Eric Nordgren's sixtieth birthday held at the University of New Hampshire in June, 1993. 1991 Mathematics Subject Classification. Primary 47A58, 47B1 5; Secondary 15A60. ? 1995 American Mathematical Society 0002-9939/95 $1.00 + $.25 per page
TL;DR: In this paper, the problem of constructing a rational matrix function which is J-unitary on the imaginary axis and has a prescribed null-pole structure in the open right half plane is solved.
Abstract: This paper presents the construction of all integral operators T on the Hilbert space L 2 l (ℝ)which appear as input-output operators of J-unitary time-varying systems and for which the set of all operators TK, where K runs over the lower triangular Hilbert-Schmidt operators on L 2 l (ℝ), is prescribed. The problem solved here may be viewed as a nonstationary analogue of the problem of constructing a rational matrix function which is J-unitary on the imaginary axis and has a prescribed null-pole structure in the open right half plane.
TL;DR: In this article, the problem of finding all completions with specific properties of a given part ia l t r iangular matrices has been studied, including the problems concerning ranks, eigenvalues, Jordan forms, singular values, as well as applications to controllability of linear systems.
Abstract: A partial triangular matrix is a matr ix in which the upper tr iangular part (including the main diagonal) is specified, and the strictly lower tr iangular part is unspecified and considered as a set of free independent variables. All matrices in this paper are over a field ~ . A completion of a part ia l mat r ix is any matr ix which is obtained by replacing the unspecified entries with elements from 9 v. A matr ix completion problem is a problem of finding all completions with specific properties of a given part ia l matrix. Various matr ix completion problems for part ia l t r iangular matrices have been studied in [1, 3, 5, 6, 7, 8, 10, 11, 12], including the problems concerning ranks, eigenvalues, Jordan forms, singular values, as well as applications to controllability of linear systems. In [1, 11] the problem of the existence of a strictly lower tr iangular completion with given characteristic polynomial of the completed matr ix has been completely solved. Generally speaking, such a completion is not unique (if exists). In [71, the possible geometric multiplicities of the eigenvalues of a completed matr ix was studied. The Jordan forms of strictly lower tr iangular completions were investigated for different part icular cases in [8, 10]. In this paper it is more convenient for us to consider strictly lower tr iangular completions as additive per turbat ions of full (not part ial) matrices. The goal of this paper is to prove a proposition describing the general sufficient condition on Jordan structures of strictly lower tr iangular additive per turbat ions of a nilpotent matrix. These conditions were conjectured by Rodman and Shalom in [10]. Given two nonincreasing sequences of positive integers {P~},\"=I and {qj}~=l, the sequence {p~}[:~ majorizes {qJ};: l if r ~ s, E~=I P~ _> E}=~ qj for t = 1 , . . . , r, and E~=~ Pl = E~=I qj (see, for example, [9]).
01 Jan 1995
TL;DR: In this paper, the construction of diagonal matrices, in some sense approximating the inverse of a given square matrix, is described, and the matrices are constructed using the well-known computer algebra system Maple.
Abstract: In this paper the construction of diagonal matrices, in some sense approximating the inverse of a given square matrix, is described. The matrices are constructed using the well-known computer algebra system Maple. The techniques we show are applicable to square matrices in general. Results are given for use in Parallel diagonal-implicit Runge-Kutta (PDIRK) methods. For an s-stage Radau IIA corrector we conjecture s! possibilities for the diagonal matrices.
07 Aug 1995
TL;DR: It is shown that the usual method for computing the coherence functions for a general multiple-input/multiple-output problem can be expressed as a modified form of Cholesky decomposition of the cross spectral density matrix of the inputs and outputs.
Abstract: It is shown that the usual method for computing the coherence functions (ordinary, partial, and multiple) for a general multiple-input/multiple-output problem can be expressed as a modified form of Cholesky decomposition of the cross spectral density matrix of the inputs and outputs. The modified form of Cholesky decomposition used is G{sub zz} = LCL{prime}, where G is the cross spectral density matrix of inputs and outputs, L is a lower; triangular matrix with ones on the diagonal, and C is a diagonal matrix, and the symbol {prime} denotes the conjugate transpose. If a diagonal element of C is zero, the off diagonal elements in the corresponding column of L are set to zero. It is shown that the results can be equivalently obtained using singular value decomposition (SVD) of G{sub zz}. The formulation as a SVD problem suggests a way to order the inputs when a natural physical order of the inputs is absent.
TL;DR: In this paper, it was shown that there exist two diagonable operators A and Z acting on l 2(Z +), that are not simultaneously similar respectivily to an upper triangular operator and a lower triangular operator.
Abstract: If A and Z are complex, square finite matrices, and if one of them is diagonable, then there exists an invertible matrix S, such that S -1 AS is upper triangular, and S -1 ZS is lower triangular. This paper presents analogues of this result for pairs of bounded operators, acting on the separable Hilbert space l 2(Z +). The main result states that there exist two diagonable operators A and Z acting on l 2(Z +), that are not simultaneously similar respectivily to an upper triangular operator and a lower triangular operator. The example is based on the existence of a unitary operator on l 2(Z +), that does not admit lower-upper factorization, even after independently permuting rows and columns. On the other hand, for pairs of bounded operators, where one of the operators is of finite rank, positive results are obtained.
TL;DR: In this article, a sufficient condition for a set of positive integers { g 1, g 2,..., g n } to be the geometric multiplicites of given eigenvalues for some strictly lower triangular completions of a partial matrix is given.
TL;DR: The study of perturbations of lower triangular partial matrices and their minimal rank completions and the results are applied to minimal representations of discrete systems.
Abstract: Questions regarding minimal representations of perturbations of discrete systems lead to the study of perturbations of lower triangular partial matrices and their minimal rank completions. Distance to the set of lower triangular partial matrices having minimal ranks smaller than a given integer is given in terms of (suitably generalized) singular numbers. Minimal ranks of lower triangular partial matrices in an arbitrary small neighborhood of a given lower triangular partial matrix are identified. The results are applied to minimal representations of discrete systems.