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Ordered multiplicity inverse eigenvalue problem for graphs on six vertices
No full textFor a graph , we associate a family of real symmetric matrices, , where for any , the location of the nonzero off-diagonal entries of is governed by the adjacency structure of . The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in . For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while with attains a particular ordered multiplicity list, it cannot do so with arbitrary spectrum
(0,1)-matrices and discrepancy
No full text Let and be positive integers, and let and be nonnegative integral vectors. Let be the set of all -matrices with row sum vector and column vector . Let and be nonincreasing, and let be the -matrix where for each , the row of consists of 1's followed by 0's. Let . The discrepancy of A, , is the number of positions in which has a 1 and has a 0. In this paper, we investigate the possible discrepancy of versus the discrepancy of . We show that if the discrepancy of is , then the discrepancy of the transpose of is at least and at most . These bounds are tight
On linear preservers of semipositive matrices
No full textGiven proper cones and in and , respectively, an matrix with real entries is said to be semipositive if there exists a such that , where denotes the interior of a proper cone . This set is denoted by . We resolve a recent conjecture on the structure of into linear preservers of . We also determine linear preservers of the set for arbitrary proper cones and . Preservers of the subclass of those elements of with a -nonnegative left inverse as well as connections between strong linear preservers of with other linear preserver problems are considered
On the estimation of for symmetric matrices
No full textThe central mathematical problem studied in this work is the estimation of the quadratic form for a given symmetric positive definite matrix and vector . Several methods to estimate without computing the matrix inverse are proposed. The precision of the estimates is analyzed both analytically and numerically.
 
Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix sequences
No full textIn recent years, motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that belongs to and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence has been identified, where is the matrix size, is the anti-identity matrix, and is the Toeplitz matrix generated by . In this note, the authors consider the multilevel Toeplitz matrix generated by , being a multi-index identifying the matrix-size, and they prove spectral and singular value distribution results for the matrix-sequence with being the corresponding tensorization of the anti-identity matrix
On the Geršgorin disks of distance matrices of graphs
No full textFor a simple connected graph , let , , , and be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix, and the distance signless Laplacian matrix of , respectively. Atik and Panigrahi [2] suggested the study of the problem: Whether all eigenvalues, except the spectral radius, of and lie in the smallest Geršgorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples
The maximal angle between 5×5 positive semidefinite and 5×5 nonnegative matrices
No full textThe paper is devoted to the study of the maximal angle between the semidefinite matrix cone and nonnegative matrix cone. A signomial geometric programming problem is formulated in the process to find the maximal angle. Instead of using an optimization problem solver to solve the problem numerically, the method of Lagrange Multipliers is used to solve the signomial geometric program, and therefore, to find the maximal angle between these two cones
Fast verification for the Perron pair of an irreducible nonnegative matrix
No full textFast algorithms are proposed for calculating error bounds for a numerically computed Perron root and vector of an irreducible nonnegative matrix. Emphasis is put on the computational efficiency of these algorithms. Error bounds for the root and vector are based on the Collatz--Wielandt theorem, and estimating a solution of a linear system whose coefficient matrix is an -matrix, respectively. We introduce a technique for obtaining better error bounds. Numerical results show properties of the algorithms
Spectral theory for self-adjoint quadratic eigenvalue problems - a review
No full textMany physical problems require the spectral analysis of quadratic matrix polynomials , , with Hermitian matrix coefficients, . In this largely expository paper, we present and discuss canonical forms for these polynomials under the action of both congruence and similarity transformations of a linearization and also -dependent unitary similarity transformations of the polynomial itself. Canonical structures for these processes are clarified, with no restrictions on eigenvalue multiplicities. Thus, we bring together two lines of attack: (a) analytic via direct reduction of the system itself by -dependent unitary similarity and (b) algebraic via reduction of symmetric linearizations of the system by either congruence (Section 4) or similarity (Sections 5 and 6) transformations which are independent of the parameter . Some new results are brought to light in the process. Complete descriptions of associated canonical structures (over and over ) are provided -- including the two cases of real symmetric coefficients and complex Hermitian coefficients. These canonical structures include the so-called sign characteristic. This notion appears in the literature with different meanings depending on the choice of canonical form. These sign characteristics are studied here and connections between them are clarified. In particular, we consider which of the linearizations reproduce the (intrinsic) signs associated with the analytic (Rellich) theory (Sections 7 and 9)
Nonparallel flat portions on the boundaries of numerical ranges of 4-by-4 nilpotent matrices
No full textThe 4-by-4 nilpotent matrices whose numerical ranges have nonparallel flat portions on their boundary that are on lines equidistant from the origin are characterized. Their numerical ranges are always symmetric about a line through the origin and all possible angles between the lines containing the flat portions are attained