Related papers: Matrix exponentials
The exponential of the triangular matrix whose entries in the diagonal at distance $n$ from the principal diagonal are all equal to the sum of the inverse of the divisors of $n$ is the triangular matrix whose entries in the diagonal at…
In this tutorial, exponentiation and factorization (decomposition) formulas are derived and discussed for common matrix operators that arise in studies of classical dynamics, linear and nonlinear optics, and special relativity. To…
In this article, we illustrate the asymptotic behaviour of exponential functions of nabla fractional calculus. For this purpose, we propose a novel matrix technique to compute these functions numerically.
Expressions are given for the exponential of a hermitian matrix, A. Replacing A by iA these are explicit formulas for the Fourier transform of exp(iA). They extend to any size matrix the previous results for the 2 X 2, 3 X 3, and 4 X 4…
We obtain formulas for the coefficients of positive and negative powers of a partial theta function.
In this paper, we introduce the new construction of fractional derivatives and integrals with respect to a function, based on a matrix approach. We believe that this is a powerful tool in both analytical and numerical calculations. We begin…
We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the…
In this paper an exponential multiplicative formula for the R-matrix is provided for the twisted affine quantum algebras.
In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued…
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to…
Apparently new expressions are given for the exponential of a hermitian matrix,A, in the 2x2,3x3,and 4x4 cases. Replacing A by iA these are explicit formulas for the Fourier transform of exp(iA).
The reciprocal Pascal matrix has entries $\binom{i+j}{j}^{-1}$. Explicit formullae for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, $q$-analogues are also presented.
We propose a diagrammatic notation for matrix differentiation. Our new notation enables us to derive formulas for matrix differentiation more easily than the usual matrix (or index) notation. We demonstrate the effectiveness of our notation…
We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but…
We give formulae for first and second derivatives of generalized eigenvalues/eigenvectors of symmetric matrices and generalized singular values/singular vectors of rectangular matrices when the matrices are linear or nonlinear functions of…
In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved.…
Some important applicative problems require the evaluation of functions $\Psi$ of large and sparse and/or \emph{localized} matrices $A$. Popular and interesting techniques for computing $\Psi(A)$ and $\Psi(A)\mathbf{v}$, where $\mathbf{v}$…
Exponential operator decompositions are an important tool in many fields of physics, for example, in quantum control, quantum computation, or condensed matter physics. In this work, we present a method for obtaining such decompositions,…
We propose a quadrature-based formula for computing the exponential function of matrices with a non-oscillatory integral on an infinite interval and an oscillatory integral on a finite interval. In the literature, existing quadrature-based…
In this work, we consider a rational approximation of the exponential function to design an algorithm for computing matrix exponential in the Hermitian case. Using partial fraction decomposition, we obtain a parallelizable method, where the…