Abstract:  Let P ∈C ^m×m and Q∈C ^n×n be Hermitian and {k +1 } -potent matrices,
i.e., P ^k+1 = P = P ^∗ , Q ^k+1 = Q = Q ^∗ , where ( · ) ∗ stands for the conjugate transpose of a
matrix. A matrix X ∈C ^m×n is called {P,Q,k +1 } -reflexive (anti-reflexive) if PXQ = X
( PXQ = −X ). In this paper, the least squares solution of the matrix equation AXB = C
subject to {P,Q,k +1 } -reflexive and anti--reflexive constraints are studied by converting
into two simpler cases: k=1 and k=2.

Key words:  Matrix equations, Potent matrix, {P,Q,k +1 } -reflexive (anti-reflexive);
Canonical correlation decomposition, Least squares solution