Abstract We describe the images of multilinear polynomials arbitrary degree evaluated on 3×3 upper triangular matrix algebra over an infinite field.

The set-intersection function gives the cardinality of the intersection of two sets. In order to obtain a lower bound for the communication complexity of this function, the rank of the corresponding characteristic function-value matrices was calculated in 4] and 5]. In this note, the rank of these matrices is determined by another method of proof, which makes use of a factorization into a produ...

We study the freeness problem for matrix semigroups. We show that the freeness problem is decidable for upper-triangular 2×2 matrices with rational entries when the products are restricted to certain bounded languages. We also show that this problem becomes undecidable for large enough matrices. The full version of this work was accepted for publication in Information and Computation. It is ava...

It is shown that a real Hankel matrix admits an approximate block diagonalization in which the successive transformation matrices are upper triangular Toeplitz matrices. The structure of this factorization was first fully discussed in [1]. This approach is extended to obtain the quotients and the remainders appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree ...

Iterative methods for solving sparse triangular systems are an attractive alternative to exact forward and backward substitution if an approximation of the solution is acceptable. On modern hardware, performance benefits are available as iterative methods allow for better parallelization. In this paper, we investigate how block-iterative triangular solves can benefit from using overlap. Because...

We determine the number of isomorphism classes elementary gradings by a finite group on an algebra upper block-triangular matrices. As consequence we prove that, for abelian G , sequence numbers E ( m ) -gradings M F × matrices with entries in field characterizes . A formula over algebraically closed field, mild restrictions its characteristic, is also provided. Finally, if group, and N that ∼ ...

For integers s,k with s 0 and k 0 , we define a class of lower triangular Toeplitz matrices U (s,k) n of type (s,k) , whose non-zero entries are the classical Horadam numbers U (a,b) i . In this paper, we derive a convolution formula containing the Horadam numbers. Using this formula, we obtain several combinatorial identities involving the Horadam numbers and the generalized Fibonacci numbers....

Recently, Jeĺınek derived that the number of self-dual interval orders of reduced size n is twice the number of row-Fishburn matrices of size n by using generating functions. In this paper, we present a bijective proof of this relation by establishing a bijection between two variations of upper-triangular matrices of nonnegative integers. Using the bijection, we provide a combinatorial proof of...