Showing papers in "Linear Algebra and its Applications in 1988"

TL;DR: In this article, it was shown that in the Frobenius norm the nearest symmetric positive semidefinite matrix to an arbitrary real matrix A is (B + H)/2, where H is the symmetric polar factor of B=(A + AT)/2.

TL;DR: In this paper, two comparison theorems for algebraic Riccati equations of the form XBR -1 B ∗ X−X(A−BR −1 C) -(A −BR − 1 C) ∗X−Q−C ∗ R -1 C = 0 were discussed.

TL;DR: In this article, an 0(&t) algorithm for computing the Kronecker structure of an arbitrary m x n pencil XE -A is presented. But the algorithm is shown to be numerically stable, because only unitary transformations are used.

TL;DR: The cone of positive semidefinite matrices which have zeros in prescribed entries is studied to obtain information and, if possible, describe the ranks of external matrices in such cones in terms of the pattern of prescribed zeros.

TL;DR: In this paper, the balanced stochastic realization is introduced as a balanced solution to the continuous time positive real equations and the dual positive real equation and the structure of the associated balanced spectral factors is derived, the main result being a product decomposition of the spectral factors.

TL;DR: In this paper, a notion of poles and zeros is developed in terms of factorizations of operator polynomials with time-varying coefficients, which is applied to the study of zero-input response and asymptotic stability.

TL;DR: In this paper, the spectral radius ϱ(A) of a simple connected graph with n vertices and m edges is defined as the adjacency matrix of the graph G and A is a matrix whose spectral radius is equal to 2m − n + 1 with equality if and only if G is isomorphic to one of the following two graphs: (a) the star K1,n−1; (b) the complete graph Kn.

TL;DR: Two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations Ax = b are considered and it is shown that when A is a monotone matrix and all the splittings are weak regular, both models lead to convergent schemes.

TL;DR: In this article, the exponential growth number of infinite graphs is studied and its relation with the essential isoperimetric number of graphs is obtained, and various inequalities involving the essential number and the spectrum of graphs are given.

TL;DR: In this article, the authors derive upper bounds for the McMillan degree of all H∞-optimal controllers associated with design problems which may be embedded in a certain generalized regular configuration.

TL;DR: In this article, methods for identifying a Lie algebra L, given by its structure constants, have been given, which lead to simple algorithms that have been implemented as computer programs, involving some symbolic manipulations.

TL;DR: In this article, bounds for the number of null elements in an eigenvector, for the multiplicity of eigenvalue and for the magnitudes of the second and last eigenvalues of a general or bipartite graph are established.

TL;DR: In this paper, Cramer's rule and Cayley-Hamilton theorem are provided in the so-called max algebra, which consists of the set of reals provided with two operations: maximization and addition.

TL;DR: Among the graphs with a prescribed number of edges, those with maximal index are determined as discussed by the authors, which confirms a conjecture of Brualdi and Hoffman, and thus confirms the existence of the maximal index conjecture.

TL;DR: In this article, the spectral decomposition of four linear mappings of centered inner-product matrices is obtained, i.e., a mapping of the linear hull of all centered inner product matrices onto the linear Hull of all the induced squared-distance matrices.

TL;DR: In this paper, a method for expanding arbitrary powers of the characteristic polynomial of a matrix is developed, expressed in terms of matrix functions generalizing those of the permanent and determinant.

TL;DR: In this paper, the authors consider systems of linear differential and algebraic equations in which some of the variables are distinguished as external variables, and give an operational form for this definition of equivalence, i.e., a set of system transformations having the property that two systems are equivalent if and only if they can be taken into each other by transformations from that set.

TL;DR: The Schur's formula is extended to the vector case, thus providing the same treatment for vector sequence transformations and the corresponding recursive algorithms, and particular rules for avoiding division by zero or numerical instability are obtained for these algorithms.

TL;DR: In this article, the problem of the existence of a p by q matrix B such that (A, B ) has prescribed controllability indices and [ λI p − A,− B ] has prescribed invariant polynomials was solved.

TL;DR: In this article, the authors survey the known forms to which a given complex matrix may be reduced by unitary or general consimilarity and describe a canonical form to which it can be reduced.

TL;DR: In this article, the authors discuss two applications of doubly stochastic matrices and related matrices: satellite-switched, time-division multiple access systems and graph automorphisms.

TL;DR: In this paper, the relationship between multivariate scattering Hurwitz polynomials and multivariate complex reactance functions is exploited to prove that a specified set of bivariate interval polynomial is characterizable by the scatteringHurwitz property from tests on a finite set of extreme bivariate polynomorphials.

TL;DR: The hyperbolic rotation algorithm is shown to be forward (weakly) stable and, in fact, comparable to an orthogonal downdating method showing to be backward stable by Stewart, and how the method's accuracy depends upon the conditioning is shown.

TL;DR: In this paper, the best approximation to a matrix by matrices positive semidefinite on a subspace is obtained, and two new characterizations of Euclidean distance matrices are presented.

TL;DR: In this article, it was shown that the dimension of a subspace of m×n matrices over any field with at least k+1 elements whose nonzero elements all have rank k is at most max(m,n).

TL;DR: In this paper, the spectral measure of a locally finite graph with bounded vertex degrees is defined, which is related to the generating function Wuv(z) for the number of walks of given length between u and v. The spectral measures are used to show the following.

TL;DR: This paper investigates the largest value of r for which the column rank and semiring rank of all m × n matrices over a given semiring are both r .

TL;DR: An estimate from below the smallest eigenvalue of the Hadamard product A ∘ C of an M -matrix A and an inverse M-matrix C is proved in this article.

TL;DR: In this paper, the decaying behavior of inverses of positive definite band matrices is analyzed for M-matrices that are in some sense close to Toeplitz matrices.

TL;DR: In this paper, the eigenvalues and vectors of a matrix A+VVT were obtained by means of the inertia of the matrix I − VT(λ − A)-1V.