Related papers: Computing Left Eigenvalues of Quaternion Matrices
The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the Rayleigh quotient iteration (RQI) for solving the right eigenpairs of…
A majority of numerical scientific computation relies heavily on handling and manipulating matrices, such as solving linear equations, finding eigenvalues and eigenvectors, and so on. Many quantum algorithms have been developed to advance…
This work presents a novel approach to compute the eigenvalues of non-Hermitian matrices using an enhanced shifted QR algorithm. The existing QR algorithms fail to converge early in the case of non-hermitian matrices, and our approach shows…
An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…
A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to…
We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put…
We review known factorization results in quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, the QR factorization. We prove there is a Schur factorization for commuting…
The eigenvalue density of a matrix plays an important role in various types of scientific computing such as electronic-structure calculations. In this paper, we propose a quantum algorithm for computing the eigenvalue density in a given…
The purpose of this effort is to investigate if the use of quaternion mathematics can be used to better model and simulate the electromagnetic fields that occur from moving electromagnetic charges. One observed deficiency with the commonly…
In this paper we introduce two definitions for algebraic and geometric multiplicities of a quaternion right eigenvalue. This definitions are equivalent to the classical ones. However, differently from the usual definitions, the notions of…
In this paper, we provide a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In this method, the columns of the quaternion matrix are orthogonalized in pairs by using…
We present a new approach to compute eigenvalues and eigenvectors of locally definite multiparameter eigenvalue problems by its signed multiindex. The method has the interpretation of a semismooth Newton method applied to certain functions…
One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the…
We present new polynomial-based methods for discrete-time quaternionic systems, highlighting how noncommutative multiplication modifies classical control approaches. Defining quaternionic polynomials via a backward-shift operator, we…
In order to approximate the integral $I(f)=\int_a^b f(x) dx$, where $f$ is a sufficiently smooth function, models for quadrature rules are developed using a given {\it panel} of $n (n\geq 2)$ equally spaced points. These models arise from…
Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Iterative random sparsification methods allow for the estimation of a single dominant eigenvalue at reduced cost by…
We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from…
As is well-known, the real quaternion division algebra $ {\cal H}$ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix algebras…
We present a new method to compute higher-order corrections to physical cross-sections, at Next-to-Leading Order and beyond. This method, based on the Loop Tree Duality, leads to locally integrable expressions in four dimensions. By…
We propose a new method for computing the eigenvalue decomposition of a dense real normal matrix $A$ through the decomposition of its skew-symmetric part. The method relies on algorithms that are known to be efficiently implemented, such as…