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Shmuel Friedland
October 2, 2013

### Abstract

A nonnegative definite hermitian m × m matrix A≠0 has increasing principal minors if det A[I] ≤ det A[J] for I⊂J, where det A[I] is the principal minor of A based on rows and columns in the set I ⊆ {1,...,m}. For m > 1 we show A has increasing principal minors if and only if A −1 exists and its diagonal entries are less or equal to 1.

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Miroslav Fiedler, Frank J. Hall
October 2, 2013

### Abstract

We study square matrices which are products of simpler factors with the property that any ordering of the factors yields a matrix cospectral with the given matrix. The results generalize those obtained previously by the authors.

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Mircea Merca
October 2, 2013

### Abstract

The n th-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n . Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.

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Geir Dahl
October 29, 2013

### Abstract

The notion of a transfer (or T -transform) is central in the theory of majorization. For instance, it lies behind the characterization of majorization in terms of doubly stochastic matrices. We introduce a new type of majorization transfer called L-transforms and prove some of its properties. Moreover, we discuss how L-transforms give a new perspective on Ryser’s algorithm for constructing (0; 1)-matrices with given row and column sums.

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Augusto Ferrante, Harald K. Wimmer
November 15, 2013

### Abstract

Let V and W be matrices of size n × pk and qm × n , respectively. A necessary and sufficient condition is given for the existence of a triple (A,B,C) such that V a k -step reachability matrix of (A,B) andW an m -step observability matrix of (A,C).

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Alexander Farrugia, John Baptist Gauci, Irene Sciriha
November 29, 2013

### Abstract

A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graph G with weighted edges and no loops. A graph associated with a n × n non–singular matrix with zero entries on the diagonal such that all its (n − 1) × (n − 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one– and two–vertex deleted subgraphs of a NSSD. It is shown that a necessary and sufficient condition for two–vertex deleted subgraphs of G and of the graph ⌈(G −1 ) associated with G −1 to remain NSSDs is that the submatrices belonging to them, derived from G and G −1 , are inverses. Moreover, an algorithm yielding what we term plain NSSDs is presented. This algorithm can be used to determine if a graph G with a terminal vertex is not a NSSD.

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N. Shayanfar, M. Hadizadeh
December 10, 2013

### Abstract

In this paper, an approach based on matrix polynomials is introduced for solving linear systems of partial differential equations. The main feature of the proposed method is the computation of the Smith canonical form of the assigned matrix polynomial to the linear system of PDEs, which leads to a reduced system. It will be shown that the reduced one is an independent system of PDEs having only one unknown in each equation. A comparison of the results for several test problems reveals that the method is very effective and convenient. The basic idea described in this paper can be employed to solve other linear functional systems.