3 Quaternion Representation 5 3.1 Axis-Angle to Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Quaternion to Axis-Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Quaternion to Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.4 Matrix to Quaternion . . . . . . . . . . ....

in this paper, we derive the necessary and sufficient conditions for the quaternion matrix equation xa=b to have the least-square bisymmetric solution and give the expression of such solution when the solvability conditions are met. futhermore, we consider the maximal and minimal inertias of the least-square bisymmetric solution to this equation. as applications, we derive sufficient and necess...

The main aim of this paper is to study quaternion matrix factorization for low-rank completion and its applications in color image processing. For the real-world images, we proposed a novel model called (LRQC), which adds total variation Tikhonov regularization factor matrices preserve spatial/temporal smoothness. Moreover, proximal alternating minimization (PAM) algorithm was tackle correspond...

In this paper, we first discuss the singular value decomposition (SVD) of a quaternion matrix and propose an algorithm to calculate the SVD of a quaternion matrix using its equivalent complex matrix. The singular values of a quaternion matrix are still real and positive, but the two unitary matrices are quaternion matrices with quaternion entries. Then, applications for color image processing b...

In the present paper‎, ‎we propose an iterative algorithm for‎ ‎solving the generalized $(P,Q)$-reflexive solution of the quaternion matrix‎ ‎equation $overset{u}{underset{l=1}{sum}}A_{l}XB_{l}+overset{v} ‎{underset{s=1}{sum}}C_{s}widetilde{X}D_{s}=F$‎. ‎By this iterative algorithm‎, ‎the solvability of the problem can be determined automatically‎. ‎When the‎ ‎matrix equation is consistent over...

in the present paper‎, ‎we propose an iterative algorithm for‎ ‎solving the generalized $(p,q)$-reflexive solution of the quaternion matrix‎ ‎equation $overset{u}{underset{l=1}{sum}}a_{l}xb_{l}+overset{v} ‎{underset{s=1}{sum}}c_{s}widetilde{x}d_{s}=f$‎. ‎by this iterative algorithm‎, ‎the solvability of the problem can be determined automatically‎. ‎when the‎ ‎matrix equation is consistent over...

We define zonal polynomials of quaternion matrix argument and deduce some important formulae of zonal polynomials and hypergeometric functions of quaternion matrix argument. As an application, we give the distributions of the largest and smallest eigenvalues of a quaternion central Wishart matrix W ∼ QW (n,Σ), respectively.

In this paper, a series of bicomplex representation methods of quaternion division algebra is introduced. We present a new multiplication concept of quaternion matrices, a new determinant concept, a new inverse concept of quaternion matrix and a new similar matrix concept. Under the new concept system, many quaternion algebra problems can be transformed into complex algebra problems to express ...

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...

The research is Supported by Chongqing University postgraduates’ Science and Innovation Fund (200801A 1 A0070266). Abstract The purpose of this paper is to discuss the inequalities for the trace of self-conjugate quaternion matrix. We present the inequality for eigenvalues and trace of self-conjugate quaternion matrices. Based on the inequality above, we obtain several inequalities for the trac...