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Representations and sign pattern of the group inverse for some block matrices
No full textLet be a complex square matrix where A is square. When BCB^{\Omega} =0, rank(BC) = rank(B) and the group inverse of \left[ \begin{array}{cc} B^{\Omega} A B^{\Omega} & 0 \\ CB^{\Omega} & 0 \right] exists, the group inverse of M exists if and only if rank(BC + A)B^{\Omega}AB^{\Omega})^{\pi}B^{\Omega}A)= rank(B). In this case, a representation of M^# in terms of the group inverse and Moore-Penrose inverse of its subblocks is given. Let A be a real matrix. The sign pattern of A is a (0,+,â)-matrix obtained from A by replacing each entry by its sign. The qualitative class of A is the set of the matrices with the same sign pattern as A, denoted by Q(A). The matrix A is called S^2GI, if the group inverse of each matrix \bar{A} in Q(A) exists and its sign pattern is independent of e A. By using the group inverse representation, a necessary and sufficient condition for a real block matrix to be an S^2GI-matrix is given
Recent results on the majorization theory of graph spectrum and topological index theory
No full textSuppose Ï = (d_1,d_2,...,d_n) and Ïâ² = (dâ²_1,dâ²_2,...,dâ²_n) are two positive non- increasing degree sequences, write Ï â³ Ïâ² if and only if Ï \neq Ïâ², \sum_{i=1}^n d_i = \sum_{i=1}^n dâ²_i, and \sum_{i=1}^j d_i ⤠\sum_{i=1}^j dâ²_i for all j = 1, 2, . . . , n. Let Ï(G) and μ(G) be the spectral radius and signless Laplacian spectral radius of G, respectively. Also let G and Gâ² be the extremal graphs with the maximal (signless Laplacian) spectral radii in the class of connected graphs with Ï and Ïâ² as their degree sequences, respectively. If Ï â³ Ïâ² can deduce that Ï(G) < Ï(Gâ²) (respectively, μ(G) < μ(Gâ²)), then it is said that the spectral radii (respectively, signless Laplacian spectral radii) of G and Gâ² satisfy the majorization theorem. This paper presents a survey to the recent results on the theory and application of the majorization theorem in graph spectrum and topological index theory
The Laplacian quadratic form and edge connectivity of a graph
No full textLet G be a simple connected graph with associated positive semidefinite integral quadratic form Q(x) = \sum (x(i) â x(j))^2, where the sum is taken over all edges ij of G. It is showed that the minimum positive value of Q(x) for x â Z_n equals the edge connectivity of G. By restricting Q(x) to x â Z_{nâ1} Ã {0}, the quadratic form becomes positive definite. It is also showed that the number of minimal disconnecting sets of edges of G equals twice the number of vectors x â Z_{nâ1} Ã{0} for which the form Q attains its minimum positive value
Invertible and regular completions of operator matrices
No full textIn this paper, for given operators A â B(X) and B â B(Y), the set of all C â B(Y,X) such that the operator matrix M_C = \left[ \begin{array}{cc} A & C \\ O & B \end{array} \right] is injective, invertible, left invertible and right invertible, is described. Answers to some open questions are given. Also, in the case when A and B are relatively regular operators, the set of all C â B(Y,X) such that M_C is regular is described. In addition, a necessary and a sufficient conditions are given for MC to be regular with the inner inverse of a certain given form
A Note on Eigenvalues Location for Trace Zero Doubly Stochastic Matrices
No full textSome results on the location of the eigenvalues of trace zero doubly stochastic matrices are provided. A result similar to that provided in [H. Perfect and L. Mirsky. Spectral properties of doublyâstochastic matrices. Monatshefte fur Mathematik, 69(1):35â57, 1965.] for doubly stochastic matrices is given
Matrices of minimum norm satisfying certain prescribed band and spectral restrictions -- an extremal characterization of the discrete periodic Laplacian
No full textThis paper has been motivated by the curiosity that the circulant matrix is the positive semidefinite, tridiagonal matrix of smallest Euclidean norm having the property that and , where and are, respectively, the vector of all s and the vector of alternating and s. It then raises the following question (minimization problem): What should be the matrix if the tridiagonal restriction is replaced by a general bandwidth ()? It is first easily shown that the solution of this problem must still be a circulant matrix. Then the determination of the first row of this circulant matrix consists in solving a least-squares problem having nonnegative variables (Nonnegative Orthant) subject to linear equations. Alternatively, this problem can be viewed as the minimization of the norm of an even function vanishing at the points of the set , and whose Fourier-transform is nonnegative, vanishes at zero, and assumes the value one at . Explicit solutions are given for the special cases of , , and . The solution for the particular case of can be physically interpreted as the vibrational mode of a ring-like chain of masses and springs in which the springs link both the nearest neighbors (with positive stiffness) and the next-nearest neighbors (with negative stiffness). The paper ends wiih a numerical illustration of the six cases ()corresponding to
Sparse spectrally arbitrary patterns
No full textWe explore combinatorial matrix patterns of order n for which some matrix entries are necessarily nonzero, some entries are zero, and some are arbitrary. In particular, we are interested in when the pattern allows any monic characteristic polynomial with real coefficients, that is, when the pattern is spectrally arbitrary. We describe some order n patterns that are spectrally arbitrary. We show that each superpattern of a sparse companion matrix pattern is spectrally arbitrary. We determine all the minimal spectrally arbitrary patterns of order 2 and 3. Finally, we demonstrate that there exist spectrally arbitrary patterns for which the nilpotent-Jacobian method fails
A matrix handling of predictions of new observations under a general random-effects model
No full textAssume that a general linear random-effects model \by = \bX\bbe + \bve is given, and new observations in the future follow the linear model \by_{\!f} = \bX_{\!f}\bbe + \bve_{\!f}. This paper shows how to establish all possible best linear unbiased predictors (BLUPs) under the general linear random-effects model with original and new observations from the original observation vector \by under a most general assumption on the covariance matrix among the random vectors \bbe, \bve and \bve_{\!f}. It utilizes a standard method of solving optimization problem in the L\"owner partial ordering on a constrained quadratic matrix-valued function, and obtains analytical expressions of the BLUPs, including those for \by_{\!f}, \bX_{\!f}\bbe and \bve_{\!f}. In particular, some fundamental equalities for the BLUPs are established under the linear random-effects model
A total positivity property of the Marchenko-Pastur Law
No full textA property of the Marchenko-Pastur measure related to total positivity is presented. The theoretical results are applied to the accurate computation of the roots of the corresponding orthogonal polynomials, an important issue in the construction of Gaussian quadrature formulas
A High-Resolution Geophysical Survey of Jenny Lake: Using Lake Sediments to Construct a Continuous Record of Tectonic Activity and Earthquake-Triggered Disturbances at Grand Teton National Park
No full textThe Teton Range, WY contains a legacy of late Cenozoic uplift and periodic Quaternary glaciations. Well-preserved fault scarps along the Teton fault displace glacier deposits from the most recent (Pinedale) glaciation and provide evidence for high fault activity during the past ~15,000 years. Observations of these scarps and previous field investigations indicate that postglacial fault offset occurred through a series of major, scarp-forming earthquakes. However, the postglacial paleoseismic record of the Teton fault remains incomplete. The goal of this project is to use lake sediments, contained in lake basins positioned on the fault, to construct a history of the timing and frequency of past earthquakes at Grand Teton National Park, and assess seismic impacts on diment erosion (e.g., landslides, debris flows, slope failures) and future hazard potential. Here, we report on multibeam sonar bathymetry and seismic reflection images from Jenny Lake, collected as part of an effort to identify glacial and tectonic landforms and to characterize infill stratigraphy. Our overarching objective is to combine these datasets with lake sediment cores from Jenny Lake and other nearby lakes to construct a continuous, accurately-dated record of past earthquakes and earthquake-generated slope failures in the Tetons