Related papers: 4 x 4 matrices in Dirac parametrization: inversion…
In the 4-path vertex cover problem, the input is an undirected graph $G$ and an integer $k$. The goal is to decide whether there is a set of vertices $S$ of size at most $k$ such that every path with 4 vertices in $G$ contains at least one…
Let $\A_0, \A_1, \ldots, \A_n$ be given square matrices of size $m$ with rational coefficients. The paper focuses on the exact computation of one point in each connected component of the real determinantal variety $\{\X \in\RR^n \: :\:…
We investigate the complex reflection group $\mathfrak{G}$ associated with the octahedral group, identified as the ninth entry in the Shephard-Todd classification. We determine all irreducible representations of $\mathfrak{G}$ and compute…
We evaluate determinants of "spiral" matrices, which are matrices in which entries are spiralling from the centre of the matrices towards the outside, with prescribed increments from one entry to the next depending on whether one moves…
We study the transference through finite index extensions of the notion of equational coherence, as well as its effective counterpart. We deduce an explicit algorithm for solving the following algorithmic problem about size two integral…
We consider the direct and inverse spectral problems for Dirac operators on $(0,1)$ with matrix-valued potentials whose entries belong to $L_p(0,1)$, $p\in[1,\infty)$. We give a complete description of the spectral data (eigenvalues and…
In this note we prove an assertion made by M. Levin in 1999: the Pascal matrix modulo 2 has the property that each of the square sub-matrices laying on the upper border or on the left border has determinants, computed in $\mathbb{Z}$, equal…
This note provides formula for determinant and inverse of r-circulant matrices with general sequences of third order. In other words, the study combines many papers in the literature.
In this paper we resolve a conjecture of Zhi-Wei Sun concerning the integrality and arithmetic structure of certain trigonometric determinants. Our approach builds on techniques developed in our previous work, where trigonometric…
We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have…
Complete spectra of the staggered Dirac operator $\Dirac$ are determined in four-dimensional $SU(2)$ gauge fields with and without dynamical fermions. An attempt is made to relate the performance of multigrid and conjugate gradient…
We compute quaisideterminants and determinants of quaternionic matrices
In this paper, firstly, by a determinant of deformed Pascal's triangle, namely the normalized Hessenberg matrix determinant, to count Dyck paths, we give another combinatorial proof of the theorems which are of Catalan numbers determinant…
Toral automorphisms, represented by unimodular integer matrices, are investigated with respect to their symmetries and reversing symmetries. We characterize the symmetry groups of GL(n,Z) matrices with simple spectrum through their…
We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct…
We describe the variety of fixed points of a unipotent operator acting on the space of matrices. We compute the determinant and the rank of a generic (symmetric, or anti-symmetric) matrix in the fixed variety, yielding information about the…
Integral identities that hold between ``desired'' and ``comparison'' solutions of the radial Dirac equations for scattering precesses are considered. Applications of these identities are discussed, particularly the determination of bounds…
A matrix $T \in \M_n(\C)$ is \emph{UECSM} if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we…
Convenient parameterizations of matrices in terms of vectors transform (certain classes of) matrix equations into covariant (hence rotation-invariant) vector equations. Certain recently introduced such parameterizations are tersely…
The image of the principal minor map for n x n-matrices is shown to be closed. In the 19th century, Nansen and Muir studied the implicitization problem of finding all relations among principal minors when n=4. We complete their partial…