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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, 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.