Expressions, as well as necessary and sufficient conditions are given for the existence of the real and pure imaginary solutions to the consistent quaternion matrix equation AXB+CY D = E. Formulas are established for the extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a solution pair X = X1 +X2i+X3j+X4k and Y = Y1+Y2i+Y3j+Y4k to this equation. Moreover, necessary and sufficient cond...

In recent years, quaternion matrix completion (QMC) based on low-rank regularization has been gradually used in image processing. Unlike (LRMC) which handles RGB images by recovering each color channel separately, QMC models retain the connection of three channels and process them as a whole. Most existing quaternion-based methods formulate (LRQMC) nuclear norm (a convex relaxation rank) minimi...

Riemannian Gaussian distributions were initially introduced as basic building blocks for learning models which aim to capture the intrinsic structure of statistical populations positive-definite matrices (here called covariance matrices). While potential applications such have attracted significant attention, a major obstacle still stands in way these applications: there seems exist no practica...

Let H be the real quaternion algebra and Hm×n denote set of all m×n matrices over H. For A?Hm×n, we by A? n×m matrix obtained applying ? entrywise to transposed AT, where is a non-standard involution A?Hn×n said ?-skew-Hermicity if A=?A?. In this paper, provide some necessary sufficient conditions for existence ?-skew-Hermitian solution system equations with four unknowns AiXi(Ai)?+BiXi+1(Bi)?=...

In the theoretical explorations and numerical computations of split quaternionic mechanics, a common extremely effective tool for study quantum mechanics field theory is quaternion equality constrained least squares (LSESQ) problem. This paper first time studies generalized singular value decomposition matrices (GSVDSQ) based on $$2\times 2$$ isomorphic representation obtains GSVDSQ theorem. ad...

We present a general framework whereby analysis of interior-point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras. In particular, such analyses are extendible to the cone of positive semidefinite Hermitian matrices with complex and quaternion entries, and to the Lorentz cone. We pro...

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...

Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as fake elliptic curves. We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explici...

Non-colliding Brownian particles in one dimension is studied. N Brownian particles start from the origin at time 0 and then they do not collide with each other until finite time T . We derive the determinantal expressions for the multitime correlation functions using the self-dual quaternion matrices. We consider the scaling limit of the infinite particles N → ∞ and the infinite time interval T...

Expressions, as well as necessary and sufficient conditions are given for the existence of the real and pure imaginary solutions to the consistent quaternion matrix equation AXB+CY D = E. Formulas are established for the extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a solution pair X = X1 +X2i+X3j+X4k and Y = Y1+Y2i+Y3j+Y4k to this equation. Moreover, necessary and sufficient cond...