Showing papers in "SIAM Journal on Matrix Analysis and Applications in 1988"
TL;DR: For real or complex matrices with elements from a standard normal distribution, the condition number should be given given a random matrix, and as mentioned in this paper showed that condition number is not the right condition number for any real matrix.
Abstract: Given a random matrix, what condition number should be expected? This paper presents a proof that for real or complex $n \times n$ matrices with elements from a standard normal distribution, the ex...
1,478 citations
TL;DR: An implementation of the generalized Schur algorithm for the superfast solution of real positive definite Toeplitz systems of order n where n = 2^
u and t is the number of positive definite systems.
Abstract: We describe an implementation of the generalized Schur algorithm for the superfast solution of real positive definite Toeplitz systems of order $n + 1$, where $n = 2^
u$. Our implementation uses t...
262 citations
TL;DR: In this paper, an important optimization problem that arises in control is to minimize the largest eigenvalue (in magnitude) of a symmetric matrix function of a given matrix function.
Abstract: An important optimization problem that arises in control is to minimize $\varphi ( x )$, the largest eigenvalue (in magnitude) of a symmetric matrix function of x. If the matrix function is affine,...
203 citations
TL;DR: In this article, the authors present an overview of the periodic Lyapunov equation, both in discrete time and in continuous time, and provide necessary and sufficient conditions for the existence and uniqueness of periodic solutions.
Abstract: This paper presents an overview of the periodic Lyapunov equation, both in discrete time and in continuous time. Together with some selected results that have recently appeared in the literature, the paper provides necessary and sufficient conditions for the existence and uniqueness of periodic solutions.
121 citations
TL;DR: In this article, it was shown that the determinant of a positive definite matrix is log-concave and that the ratio of the matrix's determinant to its principal minor is concave.
Abstract: Simple inequalities from information theory prove Hadamard's inequality and some of its gen- eralizations. It is also proven that the determinant ofa positive definite matrix is log-concave and that the ratio ofthe determinant ofthe matrix to the determinant of its principal minor g, I/Ig,- 1 is concave, establishing the concavity of minimum mean squared error in linear prediction. For Toeplitz matrices, the normalized determinant g, TM is shown to decrease with n.
86 citations
TL;DR: The authors describe the properties of nongeneric TLS problems and prove that the proposed generalization remains optimal with respect to the TLS criteria for any number of observation vectors in B if additional constraints are imposed.
Abstract: Total least squares (TLS) is one method of solving overdetermined sets of linear equations $AX \approx B$ that is appropriate when there are errors in both the observation matrix B and the data matrix A. Golub and Van Loan (G. H. Golub and C. F. Van Loan, SIAM J. Numer. Anal., 17 (1980), pp. 883–893) introduced this method into the field of numerical analysis and developed an algorithm based on the singular value decomposition. However in some TLS problems, called nongeneric, their algorithm fails to compute a finite TLS solution. This paper generalizes their TLS computations in order to solve these nongeneric TLS problems. The authors describe the properties of those problems and prove that the proposed generalization remains optimal with respect to the TLS criteria for any number of observation vectors in B if additional constraints are imposed. Finally, the TLS computation is summarized in one algorithm which includes the proposed generalization.
75 citations
TL;DR: In this article, the problem of computing the eigenvalues and vectors of a matrix H = H + D$ which is obtained from an indefinite Hermitian low-rank modification D of a Hermitians matrix H with k...
Abstract: We consider the problem of computing the eigenvalues and vectors of a matrix $\tilde H = H + D$ which is obtained from an indefinite Hermitian low rank modification D of a Hermitian matrix H with k...
70 citations
TL;DR: In this article, the consistency and solutions of the matrix equations $A\bar X - XB = C, $A \bar X \pm XA^T = C$, $A ∆ ∆ XB + XA * = C$ are characterized, and it is shown that $A *$ can be obtained from A by a consimilarity transformation using a Hermitian matrix.
Abstract: The consistency and solutions of the matrix equations $A\bar X - XB = C$, $A\bar X \pm XA^T = C$, $A\bar X \pm XA^* = C$ are characterized. As a consequence it is shown that $A^T $ (respectively, $A^* $) may be obtained from A by a consimilarity transformation using a Hermitian (respectively, symmetric) matrix.
65 citations
TL;DR: In this article, the authors considered the problem of finding a descent direction for a nonsymmetric real matrix affine function of a real parameter vector from the spectral radius of the affine functions.
Abstract: Let $A( x )$ be a nonsymmetric real matrix affine function of a real parameter vector $x \in \mathcal{R}^m $, and let $\rho ( x )$ be the spectral radius of $A( x )$. The article addresses the following question: Given $x_0 \in \mathcal{R}^m $, is $\rho ( x )$ minimized locally at $x_0 $, and, if not, is it possible to find a descent direction for $\rho ( x )$ from $x_0 $? If any of the eigenvalues of $A( x_0 )$ that achieve the maximum modulus $\rho ( x_0 )$ are multiple, this question is not trivial to answer, since the eigenvalues are not differentiable at points where they coalesce. In the symmetric case, $A( x ) = A( x )^T $ for all $x,\rho ( x )$ is convex, and the question was resolved recently by Overton following work by Fletcher and using Rockafellar’s theory of subgradients. In the nonsymmetric case $\rho ( x )$ is neither convex nor Lipschitz, and neither the theory of subgradients nor Clarke’s theory of generalized gradients is applicable. A new necessary and sufficient condition is given for...
64 citations
TL;DR: In this paper, guaranteed error bounds for controllable subspaces, uncontrollable modes, unobservable modes and invariant zeros are presented for singular matrix pencils.
Abstract: Computable, guaranteed error bounds are presented for controllable subspaces and uncontrollable modes, unobservable subspaces and unobservable modes, supremal $( A,C )$ invariant subspaces in ker D, supremal $( A,C)$ controllability subspaces in ker D, the uncontrollable modes within the supremal $( A,C )$ invariant subspace in ker D, and invariant zeros. In particular the bounds apply in the nongeneric case when the solutions are ill-posed. This is done by showing that all these features are eigenspaces and eigenvalues of certain singular matrix pencils, which means they may all be computed by a single algorithm to which a perturbation theory for general singular matrix pencils can be applied. Numerical examples are included.
53 citations
TL;DR: A new, numerically stable, structure preserving method for the discrete linear quadratic control problem with single input or single output is introduced, which is similar to Byers' method in the continuous case and faster than the general $QZ$-algorithm approach of Pappas, Laub, and Sandell.
Abstract: A new, numerically stable, structure preserving method for the discrete linear quadratic control problem with single input or single output is introduced, which is similar to Byers’ method in the continuous case and faster than the general $QZ$-algorithm approach of Pappas, Laub, and Sandell.
TL;DR: A class of transformation matrices is developed with a nonorthogonal property designed to permit the efficient deletion of data from least-squares problems, which are shown to effect deletion, or simultaneous addition and deletion, of data with much less sensitivity to rounding errors than for techniques based on normal equations.
Abstract: A class of transformation matrices, analogous to the Householder matrices, is developed with a nonorthogonal property designed to permit the efficient deletion of data from least-squares problems. These matrices, which we term hyperbolic Householder, are shown to effect deletion, or simultaneous addition and deletion, of data with much less sensitivity to rounding errors than for techniques based on normal equations. When the addition/deletion sets are large, this numerical robustness is obtained at the expense of only a modest increase in computations, and when only a relatively small fraction of the data set is modified, there is a decrease in required computations. Two applications to signal processing problems are considered. First, these transformations are used to obtain a square root algorithm for windowed recursive least-squares filtering. Second, the transformations are employed to implement the rejection of spurious data from the weight vector estimation process in an adaptive array.
TL;DR: In this paper, the Golden-Thompson inequality for the trace of matrix exponentials was derived for symmetric A and B, and a new inequality for nonnormal A was obtained as a special case.
Abstract: Several inequalities involving the trace of matrix exponentials are derived. The Golden–Thompson inequality $\operatorname{tr} e^{A + B} \leqq \operatorname{tr} e^A e^B $ for symmetric A and B is obtained as a special case along with the new inequality $\operatorname{tr} e^A e^{A^T } \leqq \operatorname{tr} e^{A + A^T } $ for nonnormal A.
TL;DR: In this paper, the general 5-diagonal Toeplitz determinant of order n was given by a simple closed expression involving Chebyshev polynomials of the second kind of order $n + 1$.
Abstract: A five-diagonal Toeplitz (5DT) determinant is defined as having zeros everywhere except in its five principal diagonals, with each principal diagonal having the same element in all positions. Thus the determinant depends on five arbitrary parameters in addition to its order. The general 5DT determinant of order n is shown to be given by a simple closed expression involving Chebyshev polynomials of the second kind of order $n + 1$. An explicit generating function for the determinants is also derived such that the nth coefficient of a power series expansion of the function is the nth-order five-diagonal Toeplitz determinant.
TL;DR: In contrast to point estimation strategies like least squares or maximum likelihood, the set-theoretic parame... as mentioned in this paper, a new algorithm is proposed for solving the set theoretic parameter estimation problem.
Abstract: A new algorithm is proposed for solving the set-theoretic parameter estimation problem. In contrast to point estimation strategies like least squares or maximum likelihood, the set-theoretic parame...
TL;DR: A new method for constructing the unimodular embedding of a polynomial matrix $P( \lambda )$ is derived, which uses a fast variant of the staircase algorithm and only requires $O( p^3 )$ operations in contrast to the methods proposed up to now.
Abstract: In this paper a new method for constructing the unimodular embedding of a polynomial matrix $P( \lambda )$ is derived. As proposed by Eising, the problem can be transformed to one of embedding a pencil, derived from the polynomial matrix $P( \lambda )$. The actual embedding of the pencil is performed here via the staircase form of this pencil, which shortcuts Eising’s construction. This then leads to a new, fast, and numerically reliable algorithm for embedding a polynomial matrix. The new method uses a fast variant of the staircase algorithm and only requires $O( p^3 )$ operations in contrast to the $O( p^4 )$ methods proposed up to now (where p is the largest dimension of the pencil). At the same time we also treat the connected problem of finding the (right) null space and (right) inverse of a polynomial matrix $P( \lambda )$.
TL;DR: An algorithm for computing the eigenvectors corresponding to the m algebraically smallest or largest eigenvalues of an symmetric matrix ${\bf A}$ is described, which is closely related to the Lanczos method, but requires a constant amount of computation at each iteration.
Abstract: An algorithm for computing the eigenvectors corresponding to the m algebraically smallest or largest eigenvalues of an $n \times n$ symmetric matrix ${\bf A}$ is described. The algorithm consists of repeated applications of the Rayleigh-Ritz procedure to a sequence of subspaces of dimension $m + 1$ which converges to the desired subspace. The method is closely related to the Lanczos method, but requires a constant amount of computation at each iteration. Applications of the algorithm include the adaptive covariance eigenstructure computation, in which the matrix ${\bf A}$ can change while the algorithm is in progress.
TL;DR: In this paper, the stability of real matrices whose undirected graphs are trees is investigated and two simple criteria are given that preclude stability throughout a sign pattern class, and those sign patterns that preclude eigenvalues with real part equal to 0.
Abstract: The principal concern of this paper is with real matrices whose undirected graphs are trees. To better understand potential stability of sign pattern classes, two simple criteria are given that preclude stability throughout a sign pattern class. In addition, those sign patterns that preclude eigenvalues with real part equal to 0 are characterized and the constant inertia within such classes is determined. Such tests may be computationally significant, as calculations with specific matrices may be subject to round off error uncertainties.
TL;DR: The homotopy method might be used to find all isolated eigenpairs for large-scale $\lambda $-matrix problems on single-instruction multiple data (SIMD) machines.
Abstract: This paper describes a homotopy method used to solve the kth-degree $\lambda $-matrix problem $( A_k \lambda ^k + A_{k - 1} \lambda ^{k - 1} + \cdots + A_1 \lambda + A_0 )x = 0$. A special homotopy equation is constructed for the case where all coefficients are general $n\times n$ complex matrices. Smooth curves connecting trivial solutions to desired eigenpairs are shown to exist. The homotopy equations maintain the nonzero structure of the underlying matrices (if there is any) and the curves correspond only to different initial values of the same ordinary differential equation. Therefore, the method might be used to find all isolated eigenpairs for large-scale $\lambda $-matrix problems on single-instruction multiple data (SIMD) machines.
TL;DR: In this paper, a numerical method for finding all eigenvalues of symmetric Toeplitz matrices is proposed, where the coefficients are the coefficients in a Laurent expansion of a rational function.
Abstract: A numerical method is proposed for finding all eigenvalues of symmetric Toeplitz matrices $T_n = ( t_{j - i} )_{ij = 1}^n $, where the $\{ t_j \}$ are the coefficients in a Laurent expansion of a rational function. Matrices of this kind occur, for example, as covariance matrices of ARMA processes. The technique rests on a representation of the characteristic polynomial as $\det ( \lambda I_n - T_n ) = W_n G_{0n} G_{1n} $ in which $G_{0n} ( \lambda ) = 0$ for the eigenvalues of $T_n $ associated with symmetric eigenvectors, $G_{1n} ( \lambda ) = 0$ for those associated with skew-symmetric eigenvectors, both functions are free of extreme variations, and both can be computed with cost independent of n. It is proposed that root finding techniques be used to compute the zeros of $G_{0n} $ and $G_{1n} $. Numerical experiments indicate that the method may be useful.
TL;DR: In this paper, the Schur complement of a partitioned operator is defined by the formula $S ( A ) = A(11) - A(12) A(22) - 1} A(21) + A(24) + 1] A(20) = 0.
Abstract: The Schur complement of a partitioned operator \[ A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \] is defined by the formula $S ( A ) = A_{11} - A_{12} A_{22}^{ - 1} A_{21} $. In finite dimensions $S ( A )$ is the unique map $c \mapsto d$ defined by the equations $A_{11} c + A_{12} y = d,A_{21} c + A_{22} y = 0$. In infinite dimensions the shorted operator of a positive operator generalizes the Schur complement; however, the above matrix equations no longer hold. We show in what sense the above equations approximately hold. Applications to infinite networks are shown.
TL;DR: A reasonable coefficient sensitivity measure for state space, recursive, finite wordlength, digital filters is the sum of the $L_2 $ norm of all first-order partial derivatives of the system function with respect to the system parameters.
Abstract: A reasonable coefficient sensitivity measure for state space, recursive, finite wordlength, digital filters is the sum of the $L_2 $ norm of all first-order partial derivatives of the system function with respect to the system parameters. This measure is actually a linear lower bound approximation to the output quantization noise power. An important feature of this measure is that it can be broken down into evaluations of ARMA auto- and cross-covariance sequences, both of which can be computed efficiently and in closed form. This efficient closed form computation is a big improvement over the computational methods used by previous researchers. Their limited methods produced only approximations to the sensitivity measure and wasted computer time (i.e., these methods are open form solutions). The direct form II sensitivity, which is shown to be approx-imately inversely proportional to the sum of products of system pole and zero distances, can, as a result, usually be reduced by the judicious placement of ad...
TL;DR: In this article, the orthogonal Procrustes problem is considered and a dual approach is proposed, where the maximal value of the objective function is approximated from above.
Abstract: This paper considers a problem of rotating m matrices toward a best least-squares fit. The problem is known as the orthogonal Procrustes problem. For $m = 2$ the solution of this problem is known and can be given in a closed form using the singular value decomposition. It appears that the general case of $m > 2$ cannot be solved explicitly and an iterative procedure is required. The authors discuss a dual approach to the Procrustes problem where the maximal value of the objective function is approximated from above. This involves minimization of the sum of k largest eigenvalues of a symmetric matrix. It will be shown that under certain conditions ensuring differentiability of the obtained function at the minimum, this method gives the global solution of the Procrustes problem.
TL;DR: In this paper, the patterns for the specified entries of a partial Toeplitz matrix (whose specified entries occur on consecutive diagonals) are characterized, which ensure that a ToEplitz partial contraction may be completed to a Toeblitz contraction.
Abstract: Those patterns for the specified entries of a partial Toeplitz matrix (whose specified entries occur on consecutive diagonals) are characterized, which ensure that a Toeplitz partial contraction may be completed to a Toeplitz contraction. The answer is rather different from that of the corresponding question without the Toeplitz condition.
TL;DR: In this article, the authors generalize the concept of centrosymmetry, and consider the properties of matrices that commute with an arbitrary permutation matrix P, the P-commutative matrices.
Abstract: Centrosymmetric matrices are matrices that commute with the permutation matrix J, the matrix with ones on its cross-diagonal. This paper generalizes the concept of centrosymmetry, and considers the properties of matrices that commute with an arbitrary permutation matrix P, the P-commutative matrices. In particular,it focuses on two related classes of matrices: inflation matrices and $ZME$-matrices. The structure of P-commutative inflators is determined, and then this is used to characterize the P-commutative $ZME$-matrices. Centrosymmetric matrices in these classes are presented as a special case.
TL;DR: In this paper, the controllability and observability matrices for the rank of a solution X are derived from an auxiliary equation of type $( * ), where the right-hand side is a rank one matrix.
Abstract: The equation \[ (*) \qquad \sum_{i,k} f_{ik} A^i XB^k = C \] is studied. The controllability matrix of $( A,C )$ and the observability matrix of $( B,C )$ yield bounds for the rank of X. If the solution X is unique it can be expressed in the form \[ X = \sum_{i,k} h_{ik} A^i CB^k . \] The coefficients $h_{ik} $ are determined by an auxiliary equation of type $( * )$, where the right-hand side is a rank one matrix.
TL;DR: In this article, the authors consider the rows of a matrix of orthogonal eigencolumns as (coordinates of) points in Euclidean space, each one corresponding to a vertex of a distance-regular graph.
Abstract: The objective of this work is to find properties of a distance-regular graph G that are expressed in the eigenvectors of its adjacency matrix. The approach is to consider the rows of a matrix of orthogonal eigencolumns as (coordinates of) points in Euclidean space, each one corresponding to a vertex of G. For the second eigenvalue, the symmetry group of the points is isomorphic to the automorphism group of G. Adjacency of vertices is related to linear dependence, linear independence, and proximity of points. Relative position of points is studied by way of the polytope that is their convex hull. Several families of examples are included.
TL;DR: In this paper, an invertible linear transformation on the real vector space of Hermitian matrices is proposed. But it is not a linear transformation that maps the class of balanced inertia (signature zero) Hermitians into itself.
Abstract: Let $H ( n )$ the $n^2 $-dimensional real vector space of Hermitian matrices. Assume n is even and greater than or equal to 4. Let T be an invertible linear transformation on $H ( n )$ that maps the class of invertible, balanced inertia (signature zero) Hermitian matrices into itself. Then for some real number $c
e 0$, and an invertible matrix $S,T ( A ) = cS^* AS$ or $T ( A ) = cS^* A^T S$, for all $A \in H( n )$. T is also classified in the case where $n=2$.
TL;DR: In this article, the authors studied matrices A that are positively diagonally equivalent to matrices B that, for given positive vectors B, map u into r, and where v maps v into c. The problem is reduced to scaling a matrix for given row sums and column sums, and applying known results for the latter.
Abstract: This article studies matrices A that are positively diagonally equivalent to matrices that, for given positive vectors $u,v,r,$ and c, map u into r, and where $A^T $ map v into c. The problem is reduced to scaling a matrix for given row sums and column sums, and applying known results for the latter. Further classifications that use these results are investigated.
TL;DR: In this article, the notion of local matrix with respect to a directed graph was introduced and it was shown that every matrix over the real or complex numbers can be represented by a local matrix.
Abstract: Let $D = ( V,E )$ be a directed graph with n vertices. We define the notion of a local matrix with respect to D and we show that every $n \times n$ matrix, over the real or complex numbers, can be ...