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Totally bipartite tridiagonal pairs
No full textThere is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of -polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite (TB). Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey--Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the -symmetric Askey--Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; and (xi) an action of the modular group associated with a TB tridiagonal pair
On the tensor rank of the 3 x 3 permanent and determinant
No full textThe tensor rank and border rank of the determinant tensor are known to be if the characteristic is not two. In characteristic two, the existing proofs of both the upper and lower bounds fail. In this paper, we show that the tensor rank remains for fields of characteristic two as well
Extreme Points of Certain Transportation Polytopes with Fixed Total Sums
No full textTransportation matrices are nonnegative matrices with given row sum vector and column sum vector . All such matrices form the convex polytope which is called a transportation polytope and its extreme points have been classified. In this article, we consider a new class of convex polytopes consisting of certain transportation polytopes satisfying that the sum of all elements is , and the row and column sum vectors are dominated componentwise by the given positive vectors and , respectively. We characterize the extreme points of . Moreover, we give the minimal term rank and maximal permanent of
The W-weighted Drazin-star matrix and its dual
No full textAfter decades studying extensively two generalized inverses, namely Moore--Penrose inverse and Drazin inverse, currently, we found immersed in a new generation of generalized inverses (core inverse, DMP inverse, etc.). The main aim of this paper is to introduce and investigate a matrix related to these new generalized inverses defined for rectangular matrices. We apply our results to the solution of linear systems
An analog of Thompson's triangle inequality in Euclidean Jordan algebras
No full textIn a recent paper [Linear Algebra Appl., 461:92--122, 2014], Tao et al. proved an analog of Thompson's triangle inequality for a simple Euclidean Jordan algebra by using a case-by-case analysis. In this short note, we provide a direct proof that is valid on any Euclidean Jordan algebras
Simple necessary conditions for Hadamard factorizability of Hurwitz polynomials
No full textIn this paper, we focus the attention on the Hadamard factorization problem for Hurwitz polynomials. We give a new necessary condition for Hadamard factorizability of Hurwitz stable polynomials of degree and show that for this condition is also sufficient. The effectiveness of the result is illustrated during construction of examples of stable polynomials that are not Hadamard factorizable
Alternating sign and sign-restricted matrices: representations and partial orders
No full textSign-restricted matrices (SRMs) are -matrices where, ignoring 0's, the signs in each column alternate beginning with a and all partial row sums are nonnegative. The most investigated of these matrices are the alternating sign matrices (ASMs), where the rows also have the alternating sign property, and all row and column sums equal 1. We introduce monotone triangles to represent SRMs and investigate some of their properties and connections to certain polytopes. We also investigate two partial orders for ASMs related to their patterns alternating cycles and show a number of combinatorial properties of these orders
The (Un)Making of a Worker Poet: The Case of Md Mukul Hossine and Migrant Worker Writings in Singapore
No full textThis article discusses the migrant worker poet Md Mukul Hossine. Showing Mukul as the representative migrant worker poet also severely restricted and complicated his process of ‘becoming’ a poet. From a Marxist standpoint, the Singaporean literati’s dismissal of Mukul reveals the predicament of being a working-class writer in today’s neoliberal market. The particular bourgeoise ‘production mode’ of working-class literature in Singapore first ‘made’, then ‘consumed’ and ultimately ‘condemned’ Mukul. First, I examine the publication process of Mukul’s poetry and its success followed by a series of problems. In the second section, I offer a close reading of Mukul’s poems understanding Mukul’s poetics and struggles as a migrant worker poet as his poetry is seldom examined in literary criticism. Finally, I argue that the representation of migrant workers writers such as Mukul is problematic due to the nature of the whole system: how they are empowered in such a context equally does harm to them. This mode again reproduces the systematic structure of power hegemony and social inequality through the field of literature
The NIEP and the positive realization problem
No full textThe nonnegative inverse eigenvalue problem is the problem of determining necessary and sufficient conditions for a multiset of complex numbers to be the spectrum of a nonnegative real matrix of size equal to the cardinality of the multiset itself. The problem is longstanding and proved to be very difficult so that several variations have been defined by considering particular classes of multisets and nonnegative real matrices. In this paper, a novel variation of the problem is proposed. This variation is motivated by a practical application in the positive realization problem, that is the problem of characterizing existence and minimality of a positive state--space representation of a given transfer function
--normal operators
No full textA new class of operators, larger than C - symmetric operators and different than normal one named C - normal operatorsis introduced. Basic properties are given. Characterizations of this operators in finite dimensional spaces using relations with conjugate normal metrices are presented. Characterizations of Toeplitz operators and composition operators as C - normal operators are given. Bunches of examples are presented