The Equivalence Canonical Forms of Two Sets of Five Quaternion Matrices With Applications

Abstract

In this paper, the equivalence canonical forms of two sets of five quaternion matrices are considered. The simultaneous decompositions of these two sets of quaternion matrices are established. Some practical algorithms for computing the decompositions are also presented. Using the equivalence canonical forms, some practical solvability conditions for two systems of generalized Sylvester-type quaternion matrix equations are derived in terms of the ranks of the coefficient matrices. The general solutions to the systems are also provided. Some numerical examples are given to illustrate the results. The proposed simultaneous decompositions are applied to construct a new framework of color watermarks embedding. In this framework, five color watermarks can be embedded into one host image simultaneously with high efficiency. Moreover, a counterexample for the existence of the equivalence canonical form of the general matrix quaternity with the same row or column numbers is presented. The results extend the existing findings on the equivalence canonical forms of general matrix arrays.