No polynomial algorithms are known for finding the coefficients of the characteristic polynomial and characteristic equation of a matrix in max-algebra. The following are proved: (1) The task of finding the max-algebraic characteristic polynomial for permutation matrices encoded using the lengths of their constituent cycles is IVP-complete. (2) The task of finding the lowest order finite term o...

In this Note we prove that the subdiscriminants of a symmetric matrix are sums of squares. This generalizes a result of [2] stating that the discriminant of a symmetric is a sum of squares and is inspired by its proof. A different, less explicit proof that the discriminant of a symmetric is a sum of squares also apear in [3]. As a consequence, we obtain an algebraic proof of the fact that all t...

This paper presents a method based on elementary transformations which may be applied to a matrix A, whose characteristic polynomial has been decomposed into linear factors, in order to obtain a nonsingular matrix P such that P-1AP is in Jordan normal form. This method can be used in the classroom, among other problems, to directly solve a linear ODE with constant coefficients. We also present ...

In this paper, We find a way to construct FSRs. The constructed FSRs can be depicted in many ways. They are just the FSRs whose characteristic polynomial can be written as g = (x0 +x1)∗f for some f . Their adjacency graphs do not contain self-loops. Further more, we can divide the vertexes in their adjacency graphs into two sets such that the edges are all between the two sets. The number of th...

In this correspondence, the characteristic polynomial and hence the linear complexity of Hall’s sextic residue sequences are determined.

Richard Stanley suggested the problem of finding the characteristic polynomial of a certain hyperplane arrangement defined by xi + xj = 0, 1, which is called the Shi threshold arrangement. We present the answer of the problem, using the finite field method.

The condition for existence of solution for 2-D singular systems is usually stated as the nonsingularity of a matrix pencil in two complex variables. The determinant of this matrix pencil equals the characteristic polynomial of the system. Although many authors assume this regularity condition to be satisfied, there has been no apparent effort towards developing computational methods to evaluat...

We obtain a precise relation between the Chern-Schwartz-MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a hypersurface, this leads to simple proofs of formulas of Dimca-Papadima and Huh for the degrees of the polar map of a homogeneous polynomial, extending these formula to any algebraically closed field of charac...

Given a field F , a scalar λ ∈ F and a matrix A ∈ F, the twisted centralizer code CF (A, λ) := {B ∈ F | AB − λBA = 0} is a linear code of length n. When A is cyclic and λ 6= 0 we prove that dimCF (A, λ) = deg(gcd(cA(t), λcA(λt))) where cA(t) denotes the characteristic polynomial of A. We also show how CF (A, λ) decomposes, and we estimate the probability that CF (A, λ) is nonzero when |F | is f...

In all polynomial zerofinding algorithms, a good convergence requires a very good initial approximation of the exact roots. The objective of the work is to study the conditions for determining the initial approximations for an iterative matrix zerofinding method. The investigation is based on the Newbery’s matrix construction which is similar to Fiedler’s construction associated with a characte...