Related papers: Cramer's rule for some quaternion matrix equations
A novel matrix approximation problem is considered herein: observations based on a few fully sampled columns and quasi-polynomial structural side information are exploited. The framework is motivated by quantum chemistry problems wherein…
New formulas for the construction of Pythagorean triples and generalizations to equations of higher powers. Application of formulas to some problems, in particular Fermat's equation with n=4.
The object of this paper is to introduce a new and fascinating method of solving large linear equations, based on Cramer's rule or Gaussian elimination but employing Sylvester's determinant identity in its computation process. In addition,…
A general framework for obtaining certain types of contracted and centrally extended algebras is presented. The whole process relies on the existence of quadratic algebras, which appear in the context of boundary integrable models.
In this paper, Ostrowski and Brauer type theorems are derived for the left and right eigenvalues of a quaternionic matrix. Generalizations of Gerschgorin type theorems are discussed for the left and the right eigenvalues of a quaternionic…
We employ the M-P inverses and ranks of quaternion matrices to establish the necessary and sufficient conditions for solving a system of the dual quaternion matrix equations $(AX, XC) = (B, D)$, along with providing an expression for its…
We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special $R_{II}$ recurrence relation. We also look into some methods for generating the nodes (which lie on…
Quadrature formulas for $\int_a^b f(x) dx$ where derivative terms need only be evaluated at $a$ and $b$ in the composite rule are identified. Error bounds are given when $f:[a,b]\to\mathbb{R}$ satisfies $f^{(n-1)}$ is absolutely continuous…
The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established (see Kou and Xia, SAPM). Some profound differences between QDEs and ODEs were observed. Also, an algorithm to evaluate the…
We describe combinatorial properties of the defining row of a circulant Hadamard matrix by exploiting its orthogonality to subsequent rows, and show how to exclude several particular forms of these matrices.
The explicit formula for the elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure for the solution of systems of linear equations is applied to error analysis. Stability conditions in terms of relative…
It is argued that the internal dynamics of a baryon, as a bound state of QCD-strings and quarks, may be captured by a theory of matrix coordinates.
By using the conjugate distribution technique of Cram\'er, we obtain some expansions of large deviation probabilities for martingales with differences satisfying the conditional Bernstein's condition. The expansions are of the same order as…
In this paper, real matrix representations of split quaternions are examined in terms of the casual character of quaternion. Then, we give De-Moivre' s formula for real matrices of timelike and spacelike split quaternions, separately.…
This note contains a new combinatorial proof of Cramer's rule based on the Gessel-Viennot-Lindstrom Lemma.
Two known computation methods and one new computation method for matrix determinant over an integral domain are discussed. For each of the methods we evaluate the computation times for different rings and show that the new method is the…
We provide first order perturbation formulas for the matrix square root (in the positive semi-definite case) and the matrix modulus (in the general case). The results are new for singular matrices, and extend previously known Fr\'{e}chet…
To each 4x4 matrix of reals another 4x4 matrix is constructed, the so-called associate matrix. This associate matrix is shown to have rank 1 and norm 1 (considered as a 16D vector) if and only if the original matrix is a 4D rotation matrix.…
The complex matrix representation for a quaternion matrix is used in this paper to find necessary and sufficient conditions for the existence of an $H$-selfadjoint $m$th root of a given $H$-selfadjoint quaternion matrix. In the process,…
We report the exact fundamental solution for Kramers equation associated to a brownian gas of charged particles, under the influence of homogeneous (spatially uniform) otherwise arbitrary, external mechanical, electrical and magnetic…