Showing papers in "Linear & Multilinear Algebra in 1995"
TL;DR: A survey of graph Laplacian analysis can be found in this article, where a graph on n vertices is represented as a graph with the n-by-n matrix L(G)−D(G)-A(G), where G is the (0, 1)-adjacency matrix of G.
Abstract: Let G be a graph on n vertices. Its Laplacian is the n-by-n matrix L(G)−D(G)−A(G), where D(G) is the diagonal matrix of vertex degrees and A(G) is the (0,1)-adjacency matrix of G. This article surveys recent results on graph Laplacians.
198 citations
TL;DR: In this paper, a formula involving sub-Pfaffians for a certain weighted sum of minors of an arbitrary given matrix was established, and applied to the generating functions of shifted tableaux.
Abstract: In this paper we establish a formula involving Pfaffians for a certain weighted sum of minors of an arbitrary given matrix. First we find a formula where the sum ranges for all columns, and secondly we obtain a formula where the sum ranges for both all rows and columns as the application of the first one. The first formula is stated in the framework of quantum matrix algebra (Matq(m,n)). These sums are weighted by "sub-Pfaffians" of any given (q)skew symmetric matrices. In the last section we provide an application of the minor summation formula to the generating functions of shifted tableaux.
84 citations
TL;DR: In this article, it was shown that the set r(A,B) of square matrices whose rows (resp. columns) are independent convex combinations of a convex combination of
Abstract: We show that the set r(A,B) (resp. c(A,B)) of square matrices whose rows (resp. columns) are independent convex combinations of
46 citations
TL;DR: Lower bounds on the bandwidth, the size of a vertex separator of general Undirected graphs, and the largest common subgraph of two undirected (weighted) graphs are obtained using a projection technique developed for the quadratic assignment problem.
Abstract: Lower bounds on the bandwidth, the size of a vertex separator of general undirected graphs, and the largest common subgraph of two undirected (weighted) graphs are obtained. The bounds are based on a projection technique developed for the quadratic assignment problem, and once more demonstrate the importance of the extreme eigenvalues of the Laplacian. They will be shown to be strict for certain classes of graphs and compare favourably to bounds already known in literature. further improvement is gained by applying nonlinear optimization methods.
44 citations
TL;DR: In this article, the authors determined all graphs with the spectrum of a distance-regular graph with at most 30 vertices, except possibly for the Taylor graph on 28 vertices.
Abstract: We determine all graphs with the spectrum of a distance-regular graph with at most 30 vertices (except possibly for the Taylor graph on 28 vertices)
42 citations
TL;DR: In this paper, the majorization inequalities between the spectrum and the main diagonal of the Laplacian matrix of a simple graph on vertices V = 1, 1, n, n with vertices with eigenvalues λ 1≥…≥λn−0 were improved.
Abstract: Let G be a simple graph on vertices V={1,…,n}, with Laplacian matrix L=L(G). suppose L has eigenvalues λ1≥…≥λn−0, and that the degree sequence of G is d 1≥…λd n. In this paper we provide an improvement to the majorization inequalities between the spectrum and the main diagonal of L. The main result has as a corollary that
31 citations
TL;DR: In this article, a finite algorithm for the (2)-generalized inverse of a matrix A is presented, which is a general framework for finite algorithms for a number of important generalized inverse of matrices.
Abstract: In this paper we establish a finite algorithm for the (2)-generalized inverse of a matrix A This is a general framework for finite algorithms for a number of important generalized inverse of matrices
28 citations
TL;DR: In this article, the matrix equation AXB+CYD=Eover a simple Artinian ring was discussed and necessary and sufficient conditions for existence of a solution were given. But the authors did not consider the representation of the solution by g-inverses.
Abstract: This paper discussed the matrix equation AXB+CYD=Eover a simple Artinian ring, obtains several necessary and sufficient conditions for existence of a solution, gives the representation of solution by g-inverses, and extends a theorem of W.E. Roth's simple Artinian rings.
27 citations
TL;DR: In this paper, the Jordan form of the tensor product (λI + A)⊗(μI + B) of invertible Jordan matrices over K is determined via an equivalent study of the nilpotent tranformation S of m × n matrices X over F where(X)S = A TX + XB.
Abstract: Let A, B denote the companion matrices of the polynomials xm ,xn over a field F of prime order p and let λ,μ be non-zero elements of an extension field K of F. The Jordan form of the tensor product (λI + A)⊗(μI + B) of invertible Jordan matrices over K is determined via an equivalent study of the nilpotent tranformation S of m × n matrices X over F where(X)S = A TX + XB. Using module-theoretic concepts a Jordan basis for S is specified recursively in terms of the representations of m and n in the scale of p, and reduction formulae for the elementary divisors of S are established.
25 citations
TL;DR: In this paper, it was shown that a symmetrized tensor space does not have an orthogonal basis consisting of standard symmetric tensors if the associated permutation group is 2-transitive.
Abstract: It is shown that a symmetrized tensor space does not have an orthogonal basis consisting of standard symmetrized tensors if the associated permutation group is 2-transitive. In particular, no such basis exists if the group is the symmetric group or the algernating group as conjectured by T.-Y. Tam and the author.
24 citations
TL;DR: Using eigenvalue interlacing and Chebyshev polynomials, this paper obtained upper bounds for the diameter of regular and bipartite biregular graphs in terms of their eigenvalues.
Abstract: Using eigenvalue interlacing and Chebyshev polynomials we find upper bounds for the diameter of regular and bipartite biregular graphs in terms of their eigenvalues. This improves results of Chung and Delorme and Sole. The same method gives upper bounds for the number of vertices at a given minimum distance from a given vertex set. These results have some applications to the covering radius of error-correcting codes.
TL;DR: In this article, it was shown that if Gaussian elimination with complete pivoting is performed on a 12 by 12 Hadamard matrix, then (1, 2,2,4,3,10/3,18/5, 4, 3,6, 6,12) must be the (absolute) pivots.
Abstract: This paper settles a conjecture by Day and Peterson that if Gaussian elimination with complete pivoting is performed on a 12 by 12 Hadamard matrix, then (1,2,2,4,3,10/3,18/5,4,3,6,6,12) must be the (absolute) pivots. Our proof uses the idea of symmetric block designs to reduce the complexity that would be found in enumerating cases. In contrast, at least 30 patterns for the absolute values of the pivots have been observed for 16 by 16 Hadamard matrices. This problem is non-trivial because row and column permutations do not preserve pivots. A naive computer search would require (12!)2 trials.
TL;DR: In this article, the generalized eigenvalue problem Ax =λBx is investigated under the conditions that A is nonnegative and irreducible, there is a nonnegative vector u such that Bu>Au, and bij ⩽ ij for all i#j.
Abstract: Motivated by economic models, the generalized eigenvalue problem Ax=λBx is investigated under the conditions that A is nonnegative and irreducible, there is a nonnegative vector u such that Bu>Au, and bij ⩽ ij for all i#j. The last two conditions are equivalent to B−A being a nonsingular M-matrix. The focus is on generalizations of the Perron-Frobenius theory, the classical theory being recovered when B is the identity matrix. These generalizations include identification of a generalized eigenvalue ρ(A,B) in the interval (0,1) with a positive eigenvector, characterizations and easily computable bounds for ρ(A,B), and localization results for all generalized eigenvalues. Dropping the condition that A is irreducible, necessary and sufficient conditions for the problem to have a solution with x≥0 are formulated in terms of basic and final classes, which are natural extensions of these concepts in the classical theory.
TL;DR: In this paper, the authors characterize the linear isometrics for the vector (p,q) norm and the induced norm on m m,n matrices over the linear space of m × n matrices.
Abstract: Let M m,n be the linear space of m×n matrices over , where For 1≤p≤∞, let be the -norm of the column vector x. Suppose 1≤p,q≤∞. Define the vector (p,q) norm on M m,n by where Aj is the jth column of A for j = 1, …n, and define the induced(p,q) norm on Mm,n by These definitions include the maximum row sum norm. maximum column sum norm, and operator norm, etc as special cases. In this paper, we characterize the linear isometrics for the vector (p,q) norm and the induced (p,q) norm on M m,n . The proofs depend on certain inequalities involving these norms and the Frobenius norm. These inequalities are of independent interest.
TL;DR: In this article, two types of irreducible stochastic matrices (tridiagonal matrices and periodic jacobi matrices) are considered as transition matrices of interval and circular random walks, and a formula for the group inverse of the associated singular M-matrix is given.
Abstract: We consider two types of irreducible stochastic matrices —tridiagonal matrices and periodic jacobi matrices —which can be viewed as transition matrices of interval and circular random walks,respcetively. For both types of matrices we give a formula for the group inverse of the associated singular M-matrix. We discuss both the sign patterns and the relative sizes of the entries in these group inverses and apply our results to give qualitative information about random walks on an interval and on a circle.
TL;DR: In this paper, it was shown that graphs whose second largest eigenvalue is less than (√5−1)/2 can be characterized by a finite collection of forbidden (induced) subgraphs.
Abstract: It is well known in spectral graph theory that all (connected) graphs except complete graphs and complete multi-partite graphs have second largest eigenvalue greater than 0. Graphs whose second larges eigenvalue does not exceed 1/3 are characterized in [2]. Some characterizations of graphs whose second largest eigenvalue does not exceed (√5−1)/2 are given in [9]. In this paper we prove that graphs whose second largest eigenvalue is less than (√5−1)/2 can be characterized by a finite collection of forbidden (induced) subgraphs.
TL;DR: The Moore-Penrose generalized inverse defines an involution in the class of real positive definite n×n matrices with rank n−1 and row-sums zero as discussed by the authors.
Abstract: The Moore-Penrose generalized inverse defines an involution in the class of real positive definite n×n matrices with rank n−1 and row-sums zero. We show that there is an analogous situation for weighted undirected graphs on n vertices and for classes (n−1)-simplices in a Euclidean (n−1)-space.
TL;DR: In this article, a multilinear ∗-polynomial identity of the algebra of n×n matrices (n≥2) over a field F of characteristic different from 2 was given.
Abstract: Let Mn (F) be the algebra of n×n matrices (n≥2) over a field F of characteristic different from 2 and let ∗ be an involution in Mn (F) In case ∗ is the transpose involution, we construct a multilinear ∗ polynomial identify of Mn (F) of degree 2n−1, P 2n−1(k 1, s 2, … s 2n−1) in one skew variable and the remaining symmetric variables of minimal degree among all ∗-polynomial identities of this type We also prove that any other multilinear ∗-polynomial identity of Mn (F) of this type of degree 2n−1 is a scalar multiple of P2n−1 In case ∗ is the symplectic involution in Mn (F), we construct a ∗-polynomial identity of Mn (F) of degree 2n−1 in skew variables T2n−1 (k 1,…,k 2n−1) and we prove that if f is a ∗-polynomial identity in skew variables for Mn (F) then deg(f)≥n+n/2
TL;DR: In this article, the authors studied the boundary ∂Wc (A) of Wc(A) and showed that the number of non-analytically smooth points is at most finite.
Abstract: Let C and A be n × n complex matrices. The C-numerical range of A is the set Wc (A) = {tr(CUAU∗) unitary}in . Given c = (c1 …cn )∊ Cn, the set Wc (A) is denoted by Wc (A) and said to be the c-numerical range in the case that C is the diagonal matrix with diagonal entries c = (c1 ,…,cn ). In this paper we study the boundary ∂Wc (A) of Wc (A). Above all, we show the following:A non-differentiable point of ∂Wc (A) is a pivot of a sector which is formed by ∂Wc (A), in the case of c =(c1 … cn ) . All differentiable points of ∂Wc (A) are classified via their degrees of smoothness. For example, there exists a case in which a C1 -smooth point of ∂W(1,0,…0) (A) is not analytically smooth. However, the number of non-analytically smooth points of ∂Wc (A) is at most finite.
TL;DR: In this article, the conjugate transpose of an n×n complex matrix A and a word in A and A′ length m was investigated and the following results were shown: 1.
Abstract: Let A’ denote the conjugate transpose of an n×n complex matrix A and let (A,A) be a word in A and A′ wilh length m The following are shown: 1.If (A, A *) or its cycle contains A2 or (A *)2 and if tr (A,A *)=tr(A * A) m/2 then A is a normal matrix; 2.If the difference of the numbers of A's and A* 's in the word is k≠0, then tr (A *) = tr(A * A)m/2 if and only if A k = (A *A) k/2. A number of consequences are also presented.
TL;DR: In this article, the authors characterize chordal supergraphs in which no 4-clique includes an added edge, the same condition that appeared in the corresponding question about positive definite completions, and characterize those graphs such that the cycle conditions on all minimal cycles imply that a partial distance matrix has a distance matrix completion.
Abstract: The Euclidean distance matrix completion problem asks when a partial distance matrix has a distance matrix completion, in the event that the graph of the specified data is chordal no additional information is needed. If the graph is not chordal, more must be known about the data. In the event the data comprises a full cycle, the additional conditions are quite simple. We characterize those graphs such that the "cycle conditions" on all minimal cycles imply that a partial distance matrix has a distance matrix completion. One description of these graphs is that they have chordal supergraphs in which no 4-clique includes an added edge, the same condition that appeared in the corresponding question about positive definite completions.
TL;DR: In this paper, a simple combinatorial rule is presented to expand the plethysm Pn [S(1 a, b)](x) of a power summetric function Pn (x) and a Schur function of hook shapeS( 1 a,b)(x), as a sum of Schur functions.
Abstract: We present a simple combinatorial rule to expand the plethysm Pn [S(1 a ,b)](x) of a power summetric function Pn (x) and a Schur function of hook shapeS(1 a ,b)(x), as a sum of Schur functions. The key ingredient of our proof is a correspondence between the circle diagrams, introduced by Chen, Garsia and Remmel [6] in their SXP algorithm to compute the Schur function expansion of Pn [S λ], and certain special rim hook and transposed special rim hook tabloids which is of interest in its own right. As an application of our rule, we drive explicit formulas for the coefficient of any Schur function of hook shape in the Schur function expansion of a plethysm of any two Schur functions of hook shape.
TL;DR: It is proved that two previously known sufficient conditions for regularity of intervals matrices are equivalent in the sense that they cover the same class of interval matrices.
Abstract: It is proved that two previously known sufficient conditions for regularity of interval matrices are equivalent in the sense that they cover the same class of interval matrices.
TL;DR: In this paper, the existence of completely controllable completion of (A1,A2 ) matrices was studied and generalization of a previous result on the same problem was shown.
Abstract: Let A1 and A2 be matrices of sizes m×m and m×n, respectively. Suppose that some of the entries under the main diagonal of A1 are unknown and all the other entries of [A 1 A 2] are constant. We study the existence of a completely controllable completion of (A1,A2 ) and generalize a previous result on the same problem.
TL;DR: In this paper, a canonical basis of R n associated with a graph G on n vertices has been defined in connection with eigenspaces and star partitions of G. The canonical star basis together with eigenvalues of G determines G to an isomorphism.
Abstract: A canonical basis of R n associated with a graph G on n vertices has been defined in [15] in connection with eigenspaces and star partitions of G. The canonical star basis together with eigenvalues of G determines G to an isomorphism. We study algorithms for finding the canonical basis and some of its variations. The emphasis is on the following three special cases; graphs with distinct eigenvalues, graphs with bounded eigenvalue multiplicities and strongly regular graphs. We show that the procedure is reduced in some parts to special cases of some well known combinatorial optimization problems, such as the maximal matching problem. the minimal cut problem, the maximal clique problem etc. This technique provides another proof of a result of L. Babai et al. [2] that isomorphism testing for graphs with bounded eigenvalue multiplicities can be performend in a polynomial time. We show that the canonical basis in strongly regular graphs is related to the graph decomposition into two strongly regular induced su...
TL;DR: In this paper, singular values and maximum rank minors of generalized inverses are studied in terms of space equivalence, and the Moore-Penrose inverse A is characterized as the {1 [-inverse of A] with minimal volume.
Abstract: Singular values and maximum rank minors of generalized inverses are studied. Proportionality of maximum rank minors is explained in terms of space equivalence. The Moore-Penrose inverse A † is characterized as the {1 [-inverse of Awith minimal volume.
TL;DR: In this article, the q-numerical range of a n×n complex matrix was shown to be the union of a finite number of algebraic arcs and every such arc lies on the boundary of an elliptical disc with eccentricity 0.
Abstract: Let A be a n×n complex matrixq be a complex number with |q|< 1. We denote by W(A:q) the compact subset {y*Ax:xeCn, yeCn x*x=1y*y=1,y*x=q of C and call it the q-numerical range of A. In this paper we show that if A is a normal matrix and |q| < 1, then the boundary of the compact convex set W(A:q) is the union of a finite number of algebraic arcs and every such arc lies on the boundary of an elliptical disc with eccentricity |q| or 0.
TL;DR: In this article, it was shown that a tensor product of linear maps of products of component factors of the domain into a selection of the factors in the range of the range can be computed by a linear functional.
Abstract: We prove that a linear map of one tensor product space to another sending decomposable tensors to decomposable tensors is essentially a tensor product of linear maps of products of component factors of the domain into a selection of the factors of the range. The product of those factors of the domain not involved in the above is collapsed via a linear functional, and those factors in the range left out of the above provide a common factor in the range. In the statement of the main theorem the flanking maps are induced by permutations of the factors of the domain and the range and they present the products in manageable form. It is assumed that the underlying field has at least five members, but the necessity of this assumption is not settled. All vector spaces are finite dimensional and the tensor products have finitely many components.