Least squares solutions to the equations AX=B,XC=D with some constraints

The least squares solutions to the equations AX=B,XC=D with some constraints such as orthogonality, symmetric orthogonality, symmetric idempotence are considered. By one time singular value decomposition or eigenvalue decomposition of the matrix product generated by the coefficient matrices A,C and right hand side matrices B,D, the constrained solutions are constructed simply. Similar strategy is applied to the equations with corresponding P-commuting constraints with given symmetric matrix P. Numerical examples that show the efficiency of the proposed methods are presented.