Solving balanced Procrustes problem with some constraints by eigenvalue decomposition

The balanced Procrustes problem with some special constraints such as symmetric orthogonality and symmetric idempotence are considered. By one time eigenvalue decomposition of the matrix product generated by the matrices AA and BB, the constrained solutions are constructed simply. Similar strategy is applied to the problem with the corresponding PP-commuting constraints with a given symmetric matrix PP. Numerical examples are presented to show the efficiency of the proposed methods.