Related papers: Matrix Roots of Eventually Positive Matrices
We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued…
We prove the (generalized) principal pivot transform is matrix monotone, in the sense of the L\"owner ordering, under minimal hypotheses. This improves on the recent results of J. E. Pascoe and R. Tully-Doyle, Monotonicity of the principal…
We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed…
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…
For any positive integer n, a new family of periodic functions in power series form and of period n is used to solve in closed form a class of polynomial equation of order n. The n roots are the values of the appropriate function from that…
It is proved that the roots of combinations of matrix polynomials with real roots can be recast as eigenvalues of combinations of real symmetric matrices, under certain hypotheses. The proof is based on recent solution of the Lax…
We propose a numerical method, based on the shift-and-invert power iteration, that answers whether a symmetric matrix is positive definite ("yes") or not ("no"). Our method uses randomization. But, it returns the correct answer with high…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
N-matrices are real $n\times n$ matrices all of whose principal minors are negative. We provide (i) an $O(2^n)$ test to detect whether or not a given matrix is an N-matrix, and (ii) a characterization of N-matrices, leading to the recursive…
We present a new algorithm to decide finiteness of matrix groups defined over a field of positive characteristic. Together with previous work for groups in zero characteristic, this provides the first complete solution of the finiteness…
It is known that there is an alternative characterization of characteristic vertices for trees with positive weights on their edges via Perron values and Perron branches. Moreover, the algebraic connectivity of a tree with positive edge…
The Leinster matrix corresponding to a finite category has entries counting the number of morphisms between objects. A first question is to know which positive integer matrices come from at least one finite category. Here, that question…
Matrices over the ring of formal power series are considered. Normal forms with respect to various sub-groups of the two-sided transformations are constructed. The construction is based on the special property of the action: it induces a…
A matrix is totally positive if all of its minors are positive. This notion of positivity coincides with the type A version of Lusztig's more general total positivity in reductive real-split algebraic groups. Since skew-symmetric matrices…
We begin by considering faithful matrix representations of elementary abelian groups in prime characteristic. The representations considered are seen to be determined up to change of bases by a single number. Studying this number leads to a…
Suppose that we are given a formal power series of many variables with coefficients in $\mathbb{R}$ (or $\mathbb{C}$) and we want to compute its $n$-th (multiplicative) root. As can be expected coefficients of the root have to satisfy a…
Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the…
We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend, and use it…
In this note, we give a necessary and sufficient condition for a matrix A in M to be finitely G-determined, where M is the ring of 2 x 2 matrices whose entries are formal power series over an infinite field, and G is a group acting on M by…
For a Lie superalgebra with Cartan matrix over a field of positive characteristic, some information about its root system in terms of the system of simple roots corresponding to the Chevalley generators is described, under certain given…