Related papers: Commutative Quaternion Matrices
The use of complexified quaternions and $i$-complex geometry in formulating the Dirac equation allows us to give interesting geometric interpretations hidden in the conventional matrix-based approach.
In this short note, we present certain generalized versions of the commutator formulas of some natural operators on manifolds, and give some applications.
In a general algebraic setting, we state some properties of commutators of reflexive admissible relations.
This paper discusses the left and right ranks of quaternion matrices with Hankel structure. While they are in general different for arbitrary quaternion matrices, we show that the left and right ranks of quaternion Hankel matrices are…
We introduce a novel approach that employs techniques from noncommutative Poisson geometry to comprehend the algebra of invariants of two $n\times n$ matrices. We entirely solve the open problem of computing the algebra of invariants of two…
We show that almost commuting real orthogonal matrices are uniformly close to exactly commuting real orthogonal matrices. We prove the same for symplectic unitary matrices. This is in contrast to the general complex case, where not all…
In this study, after introducing algebraic properties of real quaternions some characterizations of quaternionic involute-evolute curves in Q are obtained. And some results and theorems for quaternionic w-curves are given. Lastly, we…
In this study, explicit differential equations representing commutative pairs of some well-known second-order linear time-varying systems have been derived. The commutativity of these systems are investigated by considering 30 second-order…
We consider a new class of quaternionic mappings, associated with the spatial partial differential equations. We describe all mappings from this class using four analytic functions of the complex variable.
Motivated by the recent work of R. Casals on binary invariants for matrix mutation, we study the matrix congruence relation on quasi-Cartan matrices. We obtain a classification and determine normal forms modulo 4. We also establish their…
Some new identities for quantum variance and covariance involving commutators are presented, in which the density matrix and the operators are treated symmetrically. A measure of entanglement is proposed for bipartite systems, based on…
Companion matrices of the second type are characterized by properties that involve bilinear maps.
In this note, we define a Gaussian probability distribution over matrices. We prove some useful properties of this distribution, namely, the fact that marginalization, conditioning, and affine transformations preserve the matrix Gaussian…
The theory of quaternionic modular forms has been studied for decades as an example of the modular forms of many variables. The purpose of this study is to provide some congruence relations satisfied by such quaternionic modular forms.
Quaternions, discovered by Sir William Rowan Hamilton in the 19th century, are a significant extension of complex numbers and a profound tool for understanding three-dimensional rotations. This work explores the quaternion's history,…
To better understand the algebra $\mathcal{M}_n$ of all $n\times n$ complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results…
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 explore more the properties of bipartite knots and its rich combinatorial structure.
The paper aims to adopt the complex quaternion and octonion to formulate the field equations for electromagnetic and gravitational fields. Applying the octonionic representation enables one single definition to combine some physics contents…
In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions.