with given matrices
plays a very important role in matrix theory and applications and has been studied extensively. It is, in particular, connected with a certain growth curve model where it is important to be able to estimate the parameter matrix
. The authors derive the maximal and minimal ranks of the submatrices of a least squares solution matrix
and from these formulas they derive necessary and sufficient conditions for the submatrices to be 0 or other special forms. Finally, they obtain necessary and sufficient conditions for a least squares solution matrix
to be Hermitian or locally Hermitian.