Related papers: A closed-form expression for the kth power of semi…
We provide a novel recursive method, which does not require any assumption, to compute the entries of the kth power of a semicirculant matrix. As an application, a method for computing the entries of the kth power of r-circulant matrices is…
This paper shows how to obtain a simple closed form for the elements of a triangular matrix raised to the nth power.
On the Waring's problems for matrices over a commutative ring, there are some trace conditions provided for matrices eligibly expressed as a sum of $k$-th powers with $k=2,3,4,5,6,7,8$ in several literatures. In this paper, we provide the…
This note presents a simple, universal closed form for the powers of any square matrix. A diligent search of the internet gave no indication that the form is known.
This contribution is motivated by old and recent works on matrix powers and their applications on combinatorial sequences. We give in this paper the $s$-th powers and the inverses for special upper triangular matrices and the $s$-th powers…
In this work, new closed-form formulas for the matrix exponential are provided. Our method is direct and elementary, it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Moreover,…
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, closed formulas for the eigenvectors of a particular class of matrices generated by generalized permutation matrices, named generalized circulant matrices, are presented.
In this paper we deal with the problem of computing the sum of the $k$-th powers of all the elements of the matrix ring $\mathbb{M}_d(R)$ with $d>1$ and $R$ a finite commutative ring. We completely solve the problem in the case…
In this paper we compute the sum of the $k$-th powers over any finite commutative unital rings, thus generalizing known results for finite fields, the rings of integers modulo $n$ or the ring of Gaussian integers modulo $n$. As an…
We present several closed-form expressions of useful mass distributions. These include the potentials and accelerations of circular rings and arcs, the potentials of uniform density rings and arcs at arbitrary eccentricities, and the…
In recent papers, R. Bhatia, T. Jain and P. Grover obtained formulas for directional derivatives, of all orders, of the determinant, the permanent, the $m$-th compound map and the $m$-th induced power map. In this paper we generalize these…
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
The Witt ring of symmetric bilinear forms over a field has divided power operations. On the other hand, it follows from Garibaldi-Merkurjev-Serre's work on cohomological invariants that all operations on the Witt ring are essentially linear…
In this article we obtain a general polynomial identity in $k$ variables, where $k\geq 2$ is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a $k \times k$ matrix.…
In this paper an exponential multiplicative formula for the R-matrix is provided for the twisted affine quantum algebras.
We study sums of $k$-potent matrices. We show the conditions by which a complex matrix $A$ can be expressed as a sums of $k$-potent matrices. Also we obtain conditions by which a complex matrix $A$ can be expressed as a sum of finite order…
Integral Cauchy theorem is used to derive closed-form expressions of the roots of a univariate polynomial of any degree as integrals of elementary functions.
In this paper, we derive the general expression of the r-th power for some n-square complex tridiagonal matrices. Additionally, we obtain the complex factorizations of Fibonacci polynomials.
In this paper, by the tools of circulant matrices and hyperelliptic curves over finite fields, we study some arithmetic properties of certain determinants involving the Legendre symbols and $k$-th residues.