On singular value decomposition and generalized inverse of a commutative quaternion matrix and applications

By means of a complex representation of a commutative quaternion matrix, the singular value decomposition and the generalized inverse problems of a commutative quaternion matrix are studied, and the corresponding theorems and algorithms are given. In addition, based on the singular value decomposition and generalized inverse of a commutative quaternion matrix, the numerical experiments for solving the least squares problem and the color image watermarking problem are given. Numerical experiments illustrate the effectiveness and reliability of the proposed algorithms. •The singular value decomposition theorem of a commutative quaternion matrix is established.•The algebraic algorithm is given for solving the singular value decomposition problem of a commutative quaternion matrix.•The generalized inverse theorems are derived of a commutative quaternion and a commutative quaternion matrix.•The algebraic algorithm is given for solving the generalized inverse problem of a commutative quaternion matrix.•Two algebraic algorithms are given for solving the least squares problem of a commutative quaternion matrix.