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In this paper, we construct the dual-complex Gauss transformation and propose novel algorithms for LU decomposition and partial pivoting LU decomposition of dual-complex matrices. In addition, the Cholesky and LDLHdecompositions of dual-complex matrices are also obtained as special cases. Strict authentication is designed to cater to applications that have a zero-tolerance policy for any modifications or alterations made to the protected object. To increase the feasibility, we first propose the dual-complex matrix representation of color images and design a scheme for strict color image authentication by means of the dual-complex LU decomposition. Numerical experiments are provided to demonstrate the efficiency of the algorithm.

In this paper, based on semi-tensor product of matrices, a novel structure-preserving algorithm called LCstructure-preserving algorithm is applied to quaternion Householder transformations for the first time. Then by means of the above contents, we present the LCstructure-preserving methods for the QR decomposition and column pivoting QR decomposition of quaternion matrices, which make full use of high-level operations and relation of operations between quaternion matrices and their LCrepresentation matrices. The numerical examples show that compared with the ‘qr’ of the Quaternion toolbox, the LCstructure-preserving algorithm is more effective in calculating the QR decomposition of quaternion matrices.

In this paper, we put forward two efficient methods for solving the commutative quaternion Stein matrix equation X+AXB=C. By combining real representation of a commutative quaternion matrix and H-representation or generalized H-representation of special matrices, we investigate the minimal norm least squares L-structured and generalized L-structured solutions of the previous commutative quaternion matrix equation and derive their expressions. In this way, we first present a theoretical study on extending L-structured real matrices to generalized L-structured matrices, and introduce some generalized L-structured matrices. Based on them, we then discuss their applications in commutative quaternion Stein matrix equation. The algorithms only involve real operations. Consequently, it is very simple and convenient, and it can be used for all kinds of commutative quaternion matrix equation with similar problems. Furthermore, an illustrative example is provided to show the feasibility of the given methods.

In this paper, using the real representation method, we study the reduced biquaternion matrix equation AX=B. Taking advantage of the special structure of real representation of reduced biquaternion, we transform the problem of reduced biquaternion matrix into corresponding problem of real matrix. We propose the expressions of the special minimal norm least squares solution of the reduced biquaternion matrix equation AX=B, and the corresponding algorithms only perform real arithmetic. Numerical examples are provided to illustrate that our algorithm are efficient and easily understood. We also apply the minimal norm pure imaginary least squares reduced biquaternion solution to color image restoration.