Related papers: A combinatorial determinant

We study a closure operator derived from the matrix endofunctor on the category of rings with unity. We investigate the invariance of various ring-theoretic properties under this operator. A key finding is the decisive nature of this…

A recently proposed integral representation for permanents is rederived using only elementary combinatorics. For this proof the assumption that the matrix, for which the permanent is calculated, has an inverse is not necessary.

The matrix Fej\'er-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line $\mathbb{R}$. We extend a characterization to arbitrary closed semialgebraic sets $K\subseteq \mathbb{R}$ by the use of matrix…

In this paper, we consider matrices whose entries are combinatorial sequences which can be expressed in terms of a convolution of elementary and complete homogeneous symmetric functions. We establish the total positivity of these matrices…

An equivalence relation in the symmetric group, where is a positive integer has been considered. An algorithm for calculation of the number of the equivalence classes by this relation for arbitrary integer has been described.

The matrix Fej\'er-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In the previous work of the second-named author this was extended to the characterization on arbitrary closed semialgebraic sets $K$…

This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension…

Under binary matrices we mean matrices whose entries take one of two values. In this paper, explicit formulae for calculating the determinant of some type of binary Toeplitz matrices are obtained. Examples of the application of the…

We describe a class of matrices whose determinants are trivial to compute. A nice example of such a matrix is given by considering the symmetric matrix with entries {i+j choose i} (mod 2) in {0,1}, 0 <= i,j < n the binomial coefficients…

Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…

We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…

All integrals available in literature and books, that are related to Sinc(=sin x/x) function, are special cases of the general form of the integral given in the title. The evaluation of the integral is divided into two cases (i) $a$ and $b$…

A very short proof of the Fej\'er-Riesz lemma is presented in the matrix case

Let $(P,\preceq)$ be a lattice and $f$ a complex-valued function on $P$. We define meet and join matrices on two arbitrary subsets $X$ and $Y$ of $P$ by $(X,Y)_f=(f(x_i\wedge y_j))$ and $[X,Y]_f=(f(x_i\vee x_j))$ respectively. Here we…

We show that certain determinantal functions of multiple matrices, when summed over the symmetries of the cube, decompose into functions of the original matrices. These are shown to be true in complete generality; that is, no properties of…

Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo 4. We generalize this to an arbitrary (not necessarily commutative) ring of finite rank over the integers using techniques from linear algebra. Our…

In this paper, one determines the formal index and the polynomial index of a matrix linear differential operator P with coefficients in Mn(C[x]) and detAm(x) not identically zero. Then, one applies these results to give a new proof of a…

Let $\FF$ be a finite field of characteristics different from two. We show that no bijective map transforms permanent into determinant when the cardinality of $\FF$ is sufficiently large. We also give an example of non-bijective map when…

Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the…

We study the problem of determining a matrix whose $k$th multiplicative compound is a prescribed matrix~$M$. The cardinality of the set of matrices whose $k$th multiplicative compound equals~$M$ is characterized in terms of $\rank(M)$. On…