A two player game (or more correctly, a two player normal-form game) is specified by two m × n payoff matrices R and C corresponding to the row and column player respectively. Each of these matrices has m rows corresponding to the m strategies of the row player and n columns corresponding to the n strategies of the column payer. The row player picks a row i ∈ [m], and the column player picks a ...

The characteristic polynomial of the adjacency matrix of a graph is noted in connection with a quantity characterizing the topological nature of structural isomers saturated hydrocarbons [S], a set of numbers that are the same for all graphs isomorphic to the graph, and others [l]. Many properties of the characteristic polynomials of the adjacency matrices of a graph and its line graph [3] have...

In the early 1950’s, M. G. Krein characterized the entire functions that are an entry of some Nevanlinna matrix, and the pairs of entire functions that are a row of some Nevanlinna matrix. In connection with Pontryagin space versions of Krein’s theory of entire operators and de Branges’ theory of Hilbert spaces of entire functions, an indefinite analog of the Nevanlinna matrices plays a role. I...

A matrix of discrimination measures (discrimination probabilities, numerical estimates of dissimilarity, etc.) satisfies Regular Minimality (RM) if every row and every column of the matrix contains a single minimal entry, and an entry minimal in its row is minimal in its column. We derive a formula for the proportion of RM-compliant matrices among all square matrices of a given size and with no...

The LDL software package is a set of short, concise routines for factorizing symmetric positive-definite sparse matrices, with some applicability to symmetric indefinite matrices. Its primary purpose is to illustrate much of the basic theory of sparse matrix algorithms in as concise a code as possible, including an elegant method of sparse symmetric factorization that computes the factorization...

We investigate the balancing of distributed compressed storage of large sparse matrices on a massively parallel computer. For fast computation of matrix{vector and matrix{matrix products on a rectangular processor array with e cient communications along its rows and columns we require that the nonzero elements of each matrix row or column be distributed among the processors located within the s...

In this paper we describe the use of the theory of generalized polar decompositions [H. Munthe-Kaas, G. R. W. Quispel, and A. Zanna, Found. Comput. Math., 1 (2001), pp. 297–324] to approximate a matrix exponential. The algorithms presented have the property that, if Z ∈ g, a Lie algebra of matrices, then the approximation for exp(Z) resides in G, the matrix Lie group of g. This property is very...

It is proved that a certain symmetric sequence (h0, h1, . . . , hd) of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly stochastic n × n matrices, is equal to the h-vector of a simplicial polytope and hence that it satisfies the conditions of the g-theorem...

In this paper we study the class of m-row matrix compositions (m-compositions, for short), i.e., m-row matrices with nonnegative integer entries in which every column has at least one non-zero element. We provide several enumerative results, various combinatorial identities, and some combinatorial interpretations. Most of these properties are an extension to matrix compositions of the combinato...

The row by row decoupling problem (RRDP) for descriptor systems is considered using proportional state feedback and input transformation. Necessary and sufficient conditions for the solvability of the RRDP are provided. These solvability conditions can be readily verified. A constructive solution to the RRDP is given so that the desired feedback and input transformation matrices can be obtained...