Related papers: Dual Complex Adjoint Matrix: Applications in Dual …

Quaternion optimization has attracted significant interest due to its broad applications, including color face recognition, video compression, and signal processing. Despite the growing literature on quadratic and matrix quaternion…

We explain the use of dual quaternions to represent poses, twists, and wrenches.

In this paper, the consimilarity of complex matrices is generalized for the split quaternions. In this regard, coneigenvalue and coneigenvector are defined for split quaternion matrices. Also, the existence of solution to the split…

We review the general problem of finding a global rotation that transforms a given set of points and/or coordinate frames (the "test" data) into the best possible alignment with a corresponding set (the "reference" data). For 3D point data,…

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…

This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in (sturm2003cones,huang2007complex,ai2011new). The enhanced property can be used to…

We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category…

We consider a new class of non-Hermitian random matrices, namely the ones which have the form of sums of freely independent terms involving unitary matrices. To deal with them, we exploit the recently developed quaternion technique. After…

With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. The solution of total least squares problems, i.e., solving $\min_{E,r}…

In (relativistic) electronic structure methods, the quaternion matrix eigenvalue problem and the linear response (Bethe-Salpeter) eigenvalue problem for excitation energies are two frequently encountered structured eigenvalue problems.…

For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple…

In this paper, we propose a spectral framework that embeds 1D and 2D quasiperiodic Helmholtz eigenvalue problems into higher-dimensional (2D and 4D) periodic spaces via the projection method \cite{jiang2014numerical, jiang2024numerical}. To…

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…

The discretized Bethe-Salpeter eigenvalue problem arises in the Green's function evaluation in many body physics and quantum chemistry. Discretization leads to a matrix eigenvalue problem for $H \in \mathbb{C}^{2n\times 2n}$ with a…

A new method to represent and approximate rotation matrices is introduced. The method represents approximations of a rotation matrix $Q$ with linearithmic complexity, i.e. with $\frac{1}{2}n\lg(n)$ rotations over pairs of coordinates,…

Thanks to its versatility, its simplicity, and its fast convergence, ADMM is among the most widely used approaches for solving a convex problem in distributed form. However, making it running efficiently is an art that requires a fine…

This paper presents a novel quadratic projection based feature extraction framework, where a set of quadratic matrices is learned to distinguish each class from all other classes. We formulate quadratic matrix learning (QML) as a standard…

Robot-world, hand-eye calibration is the problem of determining the transformation between the robot end-effector and a camera, as well as the transformation between the robot base and the world coordinate system. This relationship has been…

Dual complex numbers can represent rigid body motion in 2D spaces. Dual complex matrices are linked with screw theory, and have potential applications in various areas. In this paper, we study low rank approximation of dual complex…

For Hermitian positive definite linear systems and eigenvalue problems, the eigCG algorithm is a memory efficient algorithm that solves the linear system and simultaneously computes some of its eigenvalues. The algorithm is based on the…