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The Hermitian Null-range of a Matrix over a Finite Field
No full textLet be a prime power. For , let be the Hermitian form of . Fix an matrix over . In this paper, it is considered the case of the set . When has coefficients in the paper studies the set . The set is the numerical range of , previously introduced in a paper by Coons, Jenkins, Knowles, Luke, and Rault (case a prime ), and by the author (arbitrary ). In this paper, it is studied in details and when . If is even, is easily described for arbitrary . If is odd, then either , or , or
Explicit Block-Structures for Block-Symmetric Fiedler-like pencils
No full textIn the last decade, there has been a continued effort to produce families of strong linearizations of a matrix polynomial , regular and singular, with good properties, such as, being companion forms, allowing the recovery of eigenvectors of a regular in an easy way, allowing the computation of the minimal indices of a singular in an easy way, etc. As a consequence of this research, families such as the family of Fiedler pencils, the family of generalized Fiedler pencils (GFP), the family of Fiedler pencils with repetition, and the family of generalized Fiedler pencils with repetition (GFPR) were constructed. In particular, one of the goals was to find in these families structured linearizations of structured matrix polynomials. For example, if a matrix polynomial is symmetric (Hermitian), it is convenient to use linearizations of that are also symmetric (Hermitian). Both the family of GFP and the family of GFPR contain block-symmetric linearizations of , which are symmetric (Hermitian) when is. Now the objective is to determine which of those structured linearizations have the best numerical properties. The main obstacle for this study is the fact that these pencils are defined implicitly as products of so-called elementary matrices. Recent papers in the literature had as a goal to provide an explicit block-structure for the pencils belonging to the family of Fiedler pencils and any of its further generalizations to solve this problem. In particular, it was shown that all GFP and GFPR, after permuting some block-rows and block-columns, belong to the family of extended block Kronecker pencils, which are defined explicitly in terms of their block-structure. Unfortunately, those permutations that transform a GFP or a GFPR into an extended block Kronecker pencil do not preserve the block-symmetric structure. Thus, in this paper, the family of block-minimal bases pencils, which is closely related to the family of extended block Kronecker pencils, and whose pencils are also defined in terms of their block-structure, is considered as a source of canonical forms for block-symmetric pencils. More precisely, four families of block-symmetric pencils which, under some generic nonsingularity conditions are block minimal bases pencils and strong linearizations of a matrix polynomial, are presented. It is shown that the block-symmetric GFP and GFPR, after some row and column permutations, belong to the union of these four families. Furthermore, it is shown that, when is a complex matrix polynomial, any block-symmetric GFP and GFPR is permutationally congruent to a pencil in some of these four families. Hence, these four families of pencils provide an alternative but explicit approach to the block-symmetric Fiedler-like pencils existing in the literature
Resolution of Conjectures related to Lights Out! and Cartesian Products
No full textLights Out! is a game played on a grid of lights, or more generally on a graph. Pressing lights on the grid allows the player to turn off neighboring lights. The goal of the game is to start with a given initial configuration of lit lights and reach a state where all lights are out. Two conjectures posed in a recently published paper about Lights Out! on Cartesian products of graphs are resolved
Falling Down, Falling Apart, and Finding Home in Reservation Blues
No full textThis essay explores journeys toward and away from ‘home’ in Sherman Alexie’s 1995 novel Reservation Blues. The textual analysis is grounded in Janet Zandy’s (1993) literal and figurative conceptions of home. Situating Alexie’s magical realism within the matrix of poverty-class, race, ethnic, and postcolonial lenses, this essay reveals a range of tragic and hopeful responses to Indigenous colonization on the Spokane Reservation
The Making of the Heiltsuk Working Class: Methodism, Time Discipline, and Capitalist Subjectivities
No full textThe Heiltsuk,1 a First Nation group in British Columbia, first encountered Europeans around the beginning of the 19th century. By the 1830s, they were thoroughly engaged in the trans-Pacific fur trade and the burgeoning commercial economy of the region. The fur trade generated considerable wealth for Heiltsuk traders, who maintained autonomy as providers of an important commodity. However, by the 1880s, many Heiltsuk were employed as wage-laborers, working at a nearby cannery, or as part of logging or commercial fishing crews. This shift to a wage-labor economy was accompanied by ideological shifts, a product of formal education and, in particular, the teachings of Methodist missionaries. Using E.P. Thompson’s study of the English working class in the early Industrial Revolution, and his concept of ‘time discipline,’ these ideological transformations are viewed as components of capitalist subjectivities
Characterizing Graphs Of Maximum Principal Ratio
No full textThe principal ratio of a connected graph, denoted γ(G), is the ratio of the maximum and minimum entries of its Perron eigenvector. Cioaba and Gregory (2007) conjectured that the graph on n vertices maximizing γ(G) is a kite graph, that is, a complete graph with a pendant path. In this paper, their conjecture is prove
A note on the matrix arithmetic-geometric mean inequality
No full textThis note proves the following inequality: If for some positive integer , then for any positive definite matrices \bA_1,\bA_2,\dots,\bA_n, the following inequality holds: \begin{equation*}\label{eq:main} \frac{1}{n^3} \, \Big\|\sum_{j_1,j_2,j_3=1}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\| \,\geq\, \frac{(n-3)!}{n!} \, \Big\|\sum_{\substack{j_1,j_2,j_3=1,\\\text{, , all distinct}}}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\|, \end{equation*} where represents the operator norm. This inequality is a special case of a recent conjecture proposed by Recht and R\'{e} (2012)
A Modified Newton method for a matrix polynomial equation arising in stochastic problem
No full textThe Newton iteration is considered for a matrix polynomial equation which arises in stochastic problem. In this paper, it is shown that the elementwise minimal nonnegative solution of the matrix polynomial equation can be obtained using Newton's method if the equation satisfies the sufficient condition, and the convergence rate of the iteration is quadratic if the solution is simple. Moreover, it is shown that the convergence rate is at least linear if the solution is non-simple, but a modified Newton method whose iteration number is less than the pure Newton iteration number can be applied. Finally, numerical experiments are given to compare the effectiveness of the modified Newton method and the standard Newton method
Perturbation results and the forward order law for the Moore-Penrose inverse of a product
No full textNew expressions are given for the Moore-Penrose inverse of a product of two complex matrices. Furthermore, an expression for (AB)\dg - B\dg A\dg for the case where or is of full rank is provided. Necessary and sufficient conditions for the forward order law for the Moore-Penrose inverse of a product to hold are established. The perturbation results presented in this paper are applied to characterize some mixed-typed reverse order laws for the Moore-Penrose inverse, as well as the reverse order law
Application of Jordan Algebra for Testing Hypotheses About Structure of Mean Vector in Model with Block Compound Symmetric Covariance Structure
No full textIn this article authors derive test for structure of mean vector in model with block compound symmetric covariance structure for two-level multivariate observations. One possible structure is so called structured mean vector when its components remain constant over sites or over time points, so that mean vector is of the form with . This hypothesis is tested against alternative of unstructured mean vector, which can change over sites or over time points