Our task here is to recall part of the theory of orders and ideals in quaternion algebras. Some of the theory makes sense in the context of B/K a quaternion algebra over a field K which is the quotient field of a Dedekind ring R. For our purposes K will always be a number field, or the completion of a number field at a finite prime, and R will be the ring of integers of K. (Nevertheless, we sha...

For scenarios of coherent underwater sources at low signal-to-noise ratio (SNR), a novel quaternion-based DOA algorithm without eigendecomposition is proposed using linear vector-hydrophone array. We construct four quaternion models by judiciously arranging the received data to fully utilize statistical information incident signals. To avoid high computational complexity caused eigenvalue decom...

In this paper, we present an isomorphism between the ring of general polynomials over a division algebra D with center F and the group ring of the free monoid with [D : F ] variables over D. Using this isomorphism, we define the characteristic polynomial of any matrix over any division algebra, i.e., a general polynomial with one variable over the algebra whose roots are precisely the left eige...

Fixing a field F of characteristic different from 2 and 3, we consider pairs (A, V ) consisting of a degree 3 central simple F -algebra A and a 3-dimensional vector subspace V of the reduced trace zero elements of A which is totally isotropic for the trace quadratic form. Each such pair gives rise to a cubic form mapping an element of V to its cube; therefore we call it a cubic pair over F . Us...

 The color image from one of the color models, for instance the RGB model, can be transformed into the quaternion algebra and be represented as one quaternion image which allows to process simultaneously of all color components of the image. The color image can be also considered in different models with transformation to the octonion space with following processing in the 8-D frequency domain...

In this study, we develop a general method to solve the linear elliptic quaternionic matrix equations by means of real representation quaternion matrices. A pseudocode for our approach that provides solution is expressed. Moreover, apply well-known Slyvester and Kalman Yakubovich over algebra.

Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split two real nonlinear systems, we derive explicit formulas for computing x2+bx+c=0 quaternion

In this paper, we present an isomorphism between the ring of general polynomials over a division algebra D with center F and the group ring of the free monoid with [D : F ] variables over D. Using this isomorphism, we define the characteristic polynomial of any matrix over any division algebra, i.e., a general polynomial with one variable over the algebra whose roots are precisely the left eige...