we derive the formulas of the maximal andminimal ranks of four real matrices $x_{1},x_{2},x_{3}$ and $x_{4}$in common solution $x=x_{1}+x_{2}i+x_{3}j+x_{4}k$ to quaternionmatrix equations $a_{1}x=c_{1},xb_{2}=c_{2},a_{3}xb_{3}=c_{3}$. asapplications, we establish necessary and sufficient conditions forthe existence of the common real and complex solutions to the matrixequations. we give the exp...

In this paper, we introduce the VQM-set and study some of its algebraic properties. Its elements are triplets consisting of a vector, a quaternion and a matrix, hence the VQM abbreviation. Operations such as addition, scalar multiplication, are defined on the VQM-set, which turn it into a vector space. A special multiplication turns a subset of the VQM-set into a group, called the VQM-group. We...

The success of applying generalized complex orthogonal designs as space-time block codes recently motivated the definition of quaternion orthogonal designs as potential building blocks for space-time-polarization block codes. Since complex orthogonal designs with no zero entries have proven important in applications, this paper offers a construction for an infinite family of restricted quaterni...

The quaternion wavelet transform is a new multiscale analysis tool. Firstly, this paper studies the standard orthogonal basis of scale space and wavelet space of quaternion wavelet transform in spatial L2 R2 , proves and presents quaternion wavelet’s scale basis function and wavelet basis function concepts in spatial scale space L2 R2;H , and studies quaternion wavelet transform structure. Fina...

Let Q2m be the generalized quaternion group of order 2 m and DN the dihedral group of order 2N . We classify the orbits in Q2m and D n pm (p prime) under the Hurwitz action. 1 The Hurwitz Action Let G be a group. For a, b ∈ G, let a = bab and a = bab. The Hurwitz action on G (n ≥ 2) is an action of the n-string braid group Bn on G . Recall that Bn is given by the presentation Bn = 〈σ1, . . . , ...

Let G be a finite, quaternion-free group with exponent e, let F be a field of characteristic zero and let x be an absolutely irreducible character of G. Suppose that a Sylow 2-subgroup of the Galois group of F(J\) over F is cyclic. It is shown that if x is not real valued, then the Schur index of x over F is odd. Let G be a finite group with exponent e and let F be a field of characteristic zer...