In this paper, we present quaternion matrix algebra techniques that can be used to process the eigen analysis of a color image. Applications of Principal Component Analysis (PCA) in image processing are numerous, and the proposed tools aim to give material for color image processing, that take into account their particular nature. For this purpose, we use the quaternion model for color images a...

By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, the expressions of the least squares η-Hermitian solution with the least norm and the expressions of the least squares η-anti-Hermitian solution with the least norm are derived for the matrix equation AXB+CXD = E over quaternions.

In this paper, real matrix representations of split quaternions are examined in terms of the casual character of quaternion. Then, we give De-Moivre’ s formula for real matrices of timelike and spacelike split quaternions, separately. Finally, we state the Euler theorem for real matrices of pure split quaternions.

In this brief note the probability density of a random real, complex and quaternion determinant is rederived using the singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices.

For random selfadjoint (real symmetric, complex Hermitian, or quaternion self-dual) Toeplitz matrices and real symmetric Hankel matrices, the existence of universal limit distributions for eigenvalues and products of several independent matrices is proved. The joint moments are the integral sums related to certain pair partitions. Our method can apply to random Hankel and Toeplitz band matrices...

We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equationsA1X C1, XB1 C2, and A3XA3 C3. Moreover, formulas of the maximal andminimal ranks of four real matrices X1, X2, X3, and X4 in solution X X1 X2i X3j X4k to the system mentioned above are derived. As app...

A 4D rotation can be decomposed into a left-isoclinic and a right-isoclinic rotation. This decomposition, known as Cayley’s factorization of 4R rotations, can be performed using Elfrinkhof-Rosen method. In this paper, we present a more straightforward alternative approach based on the fact that there is an orthogonal basis, in the sense of Hilbert-Schmidt, for the space of 4×4 real orthonormal ...