We consider the question of how to solve inverse problems of the form ex(0) = x(t) for an unkown matrix A, given measurements of x(t) at different time points. Problems of this form have applications in reverse engineering gene networks. In particular, we examine the cases where A is circulant and Toeplitz. We are also able to extend our findings of the circulant case to some generalizations of...

The use of amicable sets of eight circulant matrices and four negacyclic matrices to generate orthogonal designs is relatively new. We find all 1841 possible orthogonal designs of order 40 in three variables, using only these new techniques.

This paper studies the asymptotic behavior of eigenvalues of random abelian G-circulant matrices, that is, matrices whose structure is related to a finite abelian group G in a way that naturally generalizes the relationship between circulant matrices and cyclic groups. It is shown that, under mild conditions, when the size of the group G goes to infinity, the spectral measures of such random ma...

Estimating the condition numbers of random structured matrices is a well known challenge (cf. [SST06]), linked to the design of efficient randomized matrix algorithms in [PGMQ], [PIMR10], [PQ10], [PQ12], [PQZa], [PQa], [PQZb], [PQZC], [PY09]. We deduce such estimates for Gaussian random Toeplitz and circulant matrices. The former estimates can be surprising because the condition numbers grow ex...

We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the matrix a PGD can have at most three distinct eigenvalues, all which are nonnegative integers. The contains useful information on incidence structure design. An ordinary 2- ( v , k λ ) $(v,k,\lambda )$ design has single $\lambda $ and its circulant, geometry two...

This paper explores the relationship between Toeplitz and circulant matrices. Upper and lower bounds for all eigenvalues of hermitian Toeplitz matrices are given, capitalizing on the possibility of embedding a Toeplitz matrix in a circulant, and of expressing any nn Toeplitz matrix as a sum of two matrices with known eigenvalues. The bounds can be simultaneously found using a single discrete Fo...

In this paper, the notions of pseudofield of values and joint pseudofield of values of matrix polynomials are introduced and some of their algebraic and geometrical properties are studied.  Moreover, the relationship between the pseudofield of values of a matrix polynomial and the pseudofield of values of its companion linearization is stated, and then some properties of the augmented field of ...

1. INTRODUCTION. There is something fascinating about procedures for solving low degree polynomial equations. On one hand, we all know that while general solutions (using radicals) are impossible beyond the fourth degree, they have been found for quadratics, cubics, and quartics. On the other hand, the standard solutions for the the cubic and quartic are complicated, and the methods seem ad hoc...