When n2 elements are given, they can be arranged in the form of a square; similarly, when n8 elements (g ≥ 3 an integer) are given, they can be arranged in the form of a g-dimensional cube of side n (in short, a g-cube). The position of the elements can be indicated by g suffixes. Disciplines Physical Sciences and Mathematics Publication Details Hammer, J and Seberry, J, Higher dimensional orth...

We propose two simple upper bounds for the joint spectral radius of sets of nonnegative matrices. These bounds, the joint column radius and the joint row radius, can be computed in polynomial time as solutions of convex optimization problems. We show that for general matrices these bounds are within a factor 1/n of the exact value, where n is the size of the matrices. Moreover, for sets of matr...

It has been shown that combinatorial optimization of matrix-vector multiplication can lead to faster evaluation of finite element stiffness matrices. Based on a graph model characterizing relationships between rows, an efficient set of operations can be generated to perform matrix-vector multiplication for this problem. We improve the graph model by extending the set of binary row relationships...

This paper presents a fast incremental algorithm for low rank approximations or dimensionality reduction of matrices. Assuming that matrices have double-sided type of decomposition, we can set up an incremental solution that constitutes two coupled eigenmodels and thus a two-step updating procedure. At each step, we first represent row-row or column-column covariance matrix as the form of eigen...

A circulant matrix of order n is the matrix of convolution by a fixed element of the group algebra of the cyclic group Zn. Replacing Zn by an arbitrary finite group G gives the class of matrices that we call G-circulant. We determine the eigenvalues of such matrices with the tools of representation theory and the non-abelian Fourier transform. Definition 1 An n by n matrix C is circulant if the...

1. Results. It is well known [2] that if A and B are n and msquare matrices respectively then (1) det(^ ® B) = (det(A))m(det(B))» where A®B is the tensor or direct product of A and B. By taking absolute values on both sides of (1) we can rewrite the equality as (2) I det(4 B) |2 = (det(4^*))'»(det(73*P))«, where A* is the conjugate transpose of A. The main result is a direct extension of (2...

Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to n× n alternating sign matrices when prescribing (1,2, . . . , n) as bottom row of the array. We define monotone (d,m)-trapezoids as monotone triangles with m rows where the d − 1 top rows are removed. ...

This is a survey of the recent progress and open questions on the structure of the sets of 0-1 and non-negative integer matrices with prescribed row and column sums. We discuss cardinality estimates, the structure of a random matrix from the set, discrete versions of the Brunn-Minkowski inequality and the statistical dependence between row and column sums.