Showing papers in "Linear & Multilinear Algebra in 2005"
TL;DR: It is shown that several properties of Fiedler's definition remain valid for directed graphs and present properties peculiar to directed graphs, and inequalities relating the algebraic connectivity to quantities such as the bisection width, maximum directed cut and the isoperimetric number are proved.
Abstract: We consider a generalization of Fiedler's notion of algebraic connectivity to directed graphs. We show that several properties of Fiedler's definition remain valid for directed graphs and present properties peculiar to directed graphs. We prove inequalities relating the algebraic connectivity to quantities such as the bisection width, maximum directed cut and the isoperimetric number. Finally, we illustrate an application to the synchronization in networks of coupled chaotic systems.
163 citations
TL;DR: In this article, the general form of continuous bijective maps on n ≥ 3 that preserve commutativity in both directions has been obtained, where n is the length of the shortest path.
Abstract: We obtain the general form of continuous bijective maps on , n > 3, that preserve commutativity in both directions. We also describe all bijective maps on preserving commutativity in both directions.
75 citations
TL;DR: The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the graph as mentioned in this paper, and the energy of non-complete extended p-sum (NEPS) of the graphs can be represented as a function of the starting graph energy.
Abstract: The energy of a graph is the sum of the absolute values of the eigenvalues of the graph. We study the energy of the noncomplete extended p-sum (NEPS) of the graphs, a very general composition of the graphs in which the special case is the product of graphs. We show that the energy of the product of graphs is the product of the energy of graphs, and how this result may be used to construct arbitrarily large families of noncospectral connected graphs having the same number of vertices and the same energy. Further, unlike the product, we show that the energy of any other NEPS of the graphs cannot be represented as a function of the energy of starting graphs.
37 citations
TL;DR: In this paper, it was shown that with any finite partially ordered set P (which need not be a lattice) one can associate a matrix whose determinant factors nicely with a GCD matrix.
Abstract: We show that with any finite partially ordered set P (which need not be a lattice) one can associate a matrix whose determinant factors nicely. This was also noted by D.A. Smith, although his proof uses manipulations in the incidence algebra of P while ours is combinatorial, using nonintersecting paths in a digraph. As corollaries, we obtain new proofs for and generalizations of a number of results in the literature about GCD matrices and their relatives.
34 citations
TL;DR: In this article, the authors presented a technique for combining two matrices, an n×n matrix M and an m×m matrix B, with known spectra to create an (n+m"m"−"p") matrix N whose spectrum consists of the spectrum of the matrix M. Conditions are given when the matrix N obtained in this construction is nonnegative.
Abstract: This article presents a technique for combining two matrices, an n × n matrix M and an m × m matrix B, with known spectra to create an (n + m − p) × (n + m − p) matrix N whose spectrum consists of the spectrum of the matrix M and m − p eigenvalues of the matrix B. Conditions are given when the matrix N obtained in this construction is nonnegative. Finally, these observations are used to obtain several results on how to construct a realizable list of n + 1 complex numbers (λ1,λ2,λ3,σ) from a given realizable list of n complex numbers (c 1,c 2,σ), where c 1 is the Perron eigenvalue, c 2 is a real number and σ is a list of n − 2 complex numbers.
29 citations
TL;DR: In this article, Ravindran et al. introduced a new matrix class almost (a subclass of almost N 0-matrices which are obtained as a limit of a sequence of almost n 0 -matrices) and obtained a sufficient condition for this class to hold Q-property.
Abstract: In this article, we introduce a new matrix class almost (a subclass of almost N 0-matrices which are obtained as a limit of a sequence of almost N-matrices) and obtain a sufficient condition for this class to hold Q-property. We produce a counter example to show that an almost -matrix need not be a R 0-matrix. We also introduce another two new limiting matrix classes, namely of exact order 2, for a positive vector d and prove sufficient conditions for these classes to satisfy Q-property. Murthy et al. [Murthy, G.S.R., Parthasarathy, T. and Ravindran, G., 1993, A copositive Q-matrix which is not R 0. Mathematical Programming, 61, 131–135.] showed that Pang's conjecture ( ) is not true even when E 0 is replaced by C 0. We show that Pang's conjecture is true if E 0 is replaced by almost C 0 Finally, we present a game theoretic proof of necessary and sufficient conditions of an almost P 0-matrix satisfying Q-property.
27 citations
TL;DR: In this article, the permanents for all Hadamard matrices of orders up to and including 28 were calculated and the lowest positive value taken by the permanent in these cases was established.
Abstract: By calculating the permanents for all Hadamard matrices of orders up to and including 28 we answer a problem posed by E.T.H. Wang and a similar question asked by H. Perfect. Both questions are answered by the existence of Hadamard matrices of order 20 which do not seem to be simply related but nevertheless have the same permanent. For orders up to and including 20 we also settle several other existence questions involving permanents of (+1, −1)-matrices. Specifically, we establish the lowest positive value taken by the permanent in these cases and find matrices which have equal permanent and determinant when such a matrix exists. Our results address Conjectures 19 and 36 and Problems 5 and 7 in Minc's well known catalogue of unsolved problems on permanents. We also include a little-known proof that there exists a (+1, −1)-matrix A of order n such that per(A) = 0 if and only if n + 1 is not a power of 2.
25 citations
TL;DR: In this article, it was shown that for a matrix A of size n × m, the maps are extremals of the cone of positive maps acting on A when the rank of A equals 1 or m (in which case, necessarily m ≤ n), these extremals are exposed.
Abstract: We show that for a matrix A of size n × m, the maps are extremals of the cone of positive maps acting . When the rank of A equals 1 or m (in which case, necessarily m ≤ n), we show that these extremals are exposed.
23 citations
TL;DR: In this article, the authors obtained a property on the structure of the eigenvectors of a nonsingular unicyclic mixed graph corresponding to its least eigenvalue.
Abstract: Using the result on Fiedler vectors of a simple graph, we obtain a property on the structure of the eigenvectors of a nonsingular unicyclic mixed graph corresponding to its least eigenvalue. With the property, we get some results on minimizing and maximizing the least eigenvalue over all nonsingular unicyclic mixed graphs on n vertices with fixed girth. In particular, the graphs which minimize and maximize, respectively, the least eigenvalue are given over all such graphs with girth 3.
21 citations
TL;DR: A real or complex n×n matrix is generalized doubly stochastic if all of its row sums and column sums equal one as mentioned in this paper, where n is the number of vertices in the matrix.
Abstract: A real or complex n×n matrix is generalized doubly stochastic if all of its row sums and column sums equal one. Denote by the linear space spanned by such matrices. We study the reducibility of und...
17 citations
TL;DR: In this article, the multiplicity of the eigenvalue of a Hermitian matrix for paths with more than one vertex has been improved for some paths with multiple vertices, such as paths with two vertices.
Abstract: Let A(G) be a Hermitian matrix whose graph is a given graph G. From the interlacing theorem, it is known that , where is the multiplicity of the eigenvalue θ of A(G). In this note we improve this inequality for some paths with more than one vertex.
TL;DR: In this paper, the exponent of the primitive two-colored digraph D is the minimum value of h+k taken over all such nonnegative integers h and k with h +k>0 such that for each pair (i, j) of vertices there exists an (h, k)-walk in D from i to j.
Abstract: A two-colored digraph D is primitive if there exist nonnegative integers h and k with h+k>0 such that for each pair (i, j) of vertices there exists an (h, k)-walk in D from i to j. The exponent of the primitive two-colored digraph D is the minimum value of h+k taken over all such h and k. In this article, we consider special primitive two-colored digraphs whose uncolored digraph has n+s vertices and consist of one n-cycle and one (n − 2)-cycle. We give the bounds on the exponents, and the characterizations of the extremal two-colored digraphs.
TL;DR: For generalized and double generalized stars, a complete description of all possible eigenvalue sequences is given in this article, which is obtained by first verifying all possible multiplicity lists for these graphs which, in fact, turns out to be necessary and sufficient for determining all eigen value sequences corresponding to these graphs.
Abstract: We consider describing all possible spectra of symmetric matrices associated with certain graphs by characterizing all possible ordered multiplicity lists. For generalized and double generalized stars we provide a complete description of all possible eigenvalue sequences. This result is obtained by first verifying all possible multiplicity lists for these graphs which, in fact, turns out to be necessary and sufficient for determining all eigenvalue sequences corresponding to these graphs.
TL;DR: In this paper, the structure of Jordan homomorphisms from a commutative unital ring to an arbitrary algebra over an arbitrary set of upper triangular matrices is described. And a new proof on Jordan derivations on is obtained.
Abstract: Let be the algebra of all n × n upper triangular matrices over a commutative unital ring . We describe the structure of Jordan homomorphisms from into an arbitrary algebra over . As an application a new proof of our recent result on Jordan derivations on is obtained.
TL;DR: In this paper, the authors extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix and investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.
Abstract: We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.
TL;DR: In this paper, a defect subspace of the domain of the angular operator in terms of the Schur complement is described, and variational principles for the discrete eigenvalues in such intervals of definite type are derived.
Abstract: For a self-adjoint operator in a Krein space we construct an interval [ν, μ] outside of which the operator has only a spectrum of definite type and possesses a local spectral function. As a consequence, a spectral subspace corresponding to an interval outside [ν, μ] admits an angular operator representation. We describe a defect subspace of the domain of the angular operator in terms of the Schur complement, and we derive variational principles for the discrete eigenvalues in such intervals of definite type.
TL;DR: In this paper, the Moore-Penrose inverse for sums of matrices under rank additivity conditions is revisited and some new consequences are presented, as well as their extensions to the weighted Moore-penrose inverse.
Abstract: Some results on the Moore–Penrose inverse for sums of matrices under rank additivity conditions are revisited and some new consequences are presented. Their extensions to the weighted Moore–Penrose inverse of sums of matrices under rank additivity conditions are also considered.
TL;DR: In this article, the authors obtained the general form of orthogonality preserving automorphisms of the poset of all n'×'n upper triangular idempotent matrices over an arbitrary field.
Abstract: We obtain the general form of orthogonality preserving automorphisms of the poset of all n × n upper triangular idempotent matrices over an arbitrary field . We also give examples showing that the assumption of preserving orthogonality is essential.
TL;DR: Yimin et al. as mentioned in this paper established the definition of generalized inverse, which is the inverse of matrix A with prescribed image T and kernel S over a commutative ring R, and gave an explicit expression for over integral domains.
Abstract: In this article, we establish the definition of the generalized inverse , which is {2} inverse of matrix A with prescribed image T and kernel S over a commutative ring R, and give an explicit expression for over integral domains, which generalizes the explicit expression for over the field of complex numbers [Wei Yimin, 1998, A characterization and representation of the generalized inverse and its applications, Linear Algebra Applications, 280, 87–96, Theorem 2.1]. In addition, we show that over integral domains, the Drazin inverse, the group inverse and the Moore–Penrose inverse are all .
TL;DR: In this paper, the structure of all additive rank-one preservers from Mmn (F) to Mpq (F), where m, n, p, q are positive integers, is described.
Abstract: Suppose F is a field and m, n, p, q are positive integers. Let Mmn (F) be the set of all m × n matrices over F, and let Mmn 1(F) be its subset consisting of all rank-one matrices. A map ϕ : Mmn (F) → Mpq (F) is said to be an additive rank-one preserver if and ϕ(A + B) = ϕ(A) + ϕ(B) for any A, B ∈ Mmn (F). This article describes the structure of all additive rank-one preservers from Mmn (F) to Mpq (F).
TL;DR: In this article, it was shown that if A and B are inverse M-matrices such that B −1 ≤ A −1, then (B+tI)−1 ≤ (A+TI)-−1, for all t ≥ 0.
Abstract: In earlier works, authors such as Varga, Micchelli and Willoughby, Ando, and Fiedler and Schneider have studied and characterized functions which preserve the M-matrices or some subclasses of the M-matrices, such as the Stieltjes matrices. Here we characterize functions which either preserve the inverse M-matrices or map the inverse M-matrices to the M-matrices. In one of our results we employ the theory of Pick functions to show that if A and B are inverse M-matrices such that B −1 ≤ A −1, then (B+tI)−1 ≤ (A+tI)−1, for all t ≥ 0.
TL;DR: In this article, the determinant and the inverse of the matrix [Ψ (x i,x j )], where S={x 1,x 2, …,x n } is a meet-closed or lower-closed subset of P and h∈ L S, are defined.
Abstract: Let P=(P,≤, ∧) be a meet-semilattice with least element 0 such that every principal order ideal is finite. We define the function Ψ on P × P by where f , g and h are incidence functions of P. We calculate the determinant and the inverse of the matrix [Ψ (x i ,x j )], where S={x 1,x 2, …,x n } is a meet-closed or lower-closed subset of P and h∈ L S . Here LS is the class of incidence functions defined by We apply the results to meet matrices and obtain known determinant formulae and new inverse formulae for them. These results also concern GCD matrices, which are number-theoretic special cases of meet matrices. We also apply the results to the matrix [C(x i ,x j )], where C(m, n) is the usual Ramanujan's sum.
TL;DR: In this paper, the q-numerical range of a bounded linear operator on a complex Hilbert space H with inner product ⟨·, ·⟩ is defined for a real number q ∈ [0, 1] and the fundamental properties of the function h on conv(σ (T)) defined by these fundamental properties are obtained.
Abstract: Let T be a bounded linear operator on a complex Hilbert space H with inner product ⟨·, ·⟩. For a real number q ∈ [0, 1], the q-numerical range of T is defined by We obtain fundamental properties of the function h on conv(σ (T)) defined by If T is normal, these fundamental properties are applicable to show that Furthermore, if T is a non-essential hermitian then closure(Fq (T)) is the union of Fq (diag(λ1, λ2, λ3)) over distinct extreme points λ1, λ2, λ3 of conv(σ (T)).
TL;DR: In this article, the authors characterize linear preservers of rank-sum-maximal on symmetric matrices over a field F and linear presers of rank (respectively, rank-subtractivity and rank-minimum) on rank-minimal and ranksubtractive matrices.
Abstract: Let Sn (F) be the set of all n × n symmetric matrices over a field F. For a matrix denotes the rank of A. A pair of n × n matrices ( A, B ) is said to be rank-sum-maximal if , rank-sum-minimal if and rank-subtractive if . We say that a linear operator ϕ from Sn (F) to itself is a linear preserver of rank-sum-maximum (respectively, rank-sum-minimum and rank-subtractivity) on Sn (F) if it preserves the set of all rank-sum-maximal (respectively, rank-sum-minimal and rank-subtractive) pairs, and of rank on Sn (F) if ρ(ϕ(X)) = ρ(X) for every . We first characterize the linear preservers of rank-sum-maximum on Sn (F) when F is arbitrary, and thereby, linear preservers of rank (respectively, rank-subtractivity and rank-sum-minimum) on Sn (F) are characterized.
TL;DR: A set of rank equalities and inequalities for block matrices consisting of Kronecker products are established in this article, and the consequences of these inequalities are also given in detail.
Abstract: A set of rank equalities and inequalities are established for block matrices consisting of Kronecker products. Various consequences are also given.
TL;DR: For a Hermitian positive definite non-diagonal matrix A, whose graph is a tree, this paper showed that P μ (A) is a strictly increasing function of μ ∈ [−1, 1] for a Hermanian positive- definite noniagonal matrix, where σ is a permutation permutation.
Abstract: Let A=(a ij ) be an n-by-n matrix. For any real μ, define the polynomial where l (σ) is the number of inversions of the permutation σ in the symmetric group Sn . We prove that P μ (A) is a strictly increasing function of μ ∈ [−1,1], for a Hermitian positive definite nondiagonal matrix A, whose graph is a tree.
TL;DR: In this article, the authors describe the structure of irreducible matrix groups with submultiplicative spectrum and obtain a block-monomial structure of matrices in such groups and give an upper bound to the exponent of these groups.
Abstract: We describe the structure of irreducible matrix groups with submultiplicative spectrum. Since all such groups are nilpotent, the study is focused on p-groups. We obtain a block-monomial structure of matrices in irreducible p-groups and build polycyclic series arising from that structure. We give an upper bound to the exponent of these groups. We determine all minimal irreducible groups of p× p matrices with submultiplicative spectrum and discuss the case of p 2× p 2 matrices if p is an odd prime.
TL;DR: In this paper, the possible number of ones in a 0-1 matrix with given rank in the generic case and in the symmetric case was determined in both the generic and symmetric cases.
Abstract: We determine the possible numbers of ones in a 0–1 matrix with given rank in the generic case and in the symmetric case. There are some unexpected phenomena. The rank 2 symmetric case is subtle.
TL;DR: In this paper, it was shown that an irreducible semigroup such that the spectra of all the elements of are contained in F is conjugate to a subsemigroup of Mn (F).
Abstract: Let K be an algebraically closed field of characteristic zero and F
TL;DR: Additive mappings which do not increase the minimal rank of alternate matrices are completely classified in this paper, and no condition is imposed on the underlying field, which is the case in this paper.
Abstract: Additive mappings, which do not increase the minimal rank of alternate matrices, are completely classified. No condition is imposed on the underlying field.