Eigenvalues and Singular Value Decompositions of Reduced Biquaternion Matrices

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, the algorithms for calculating the eigenvalues, the eigenvectors, and the singular value decompositions (SVD) of a reduced biquaternion (RB) matrix are developed. We use the SVD to approximate an RB matrix in the least square sense and define the pseudoinverse matrix of an RB matrix. Moreover, the RB SVD is employed to implement the SVD of a color image. The computational complexity for the SVD of an RB matrix is only one-fourth of that for the SVD of conventional quaternion matrices. Therefore, many useful image-processing methods using the SVD can be extended to a color image without separating the color image into three channels. The numbers of the eigenvalues of an <formula formulatype="inline"><tex>$n\times n$</tex></formula> RB matrix, the <formula formulatype="inline"><tex>$n^{\rm th}$</tex></formula> roots of an RB, and the zeros of an RB polynomial with degree <formula formulatype="inline"><tex>$n$</tex> </formula> are all finite and equal to <formula formulatype="inline"><tex>$n^{2}$</tex> </formula>, not infinite as those of conventional quaternions. </para>