The linear hypotheses and constraints

of the n dimensional Euclidean space Em and e is an n x 1 vector of random errors which has the multivariate normal distribution with mean 0 and covariance matrix (J2In, (J 2 times the unit matrix In. It is worthwhile to note that in our unified treatment no restriction is imposed on the known n x m matrix A and the known Ixm matrix B. The matrix A may be called a design matrix. The matrix equation Bτ = 0 is a set of constraints imposed on the parameter vector r. Bτ — 0 is in some cases a set of identifiability constraints of the parameter vector r, a set of hypotheses to be tested and a set of more complex constraints. The matrices A and B and the parameter vector τ jointly specify the linear subspace Ξ of E». The least squares estimate of the parameter τ in the extended sense and the projection operator to the space Ξ obtained by using the generalized inverse matrices are given in the Theorem of section 2. Some properties of the generalized inverse matrices and the projection operators are also given in section 2. Our general formula given in the Theorem contains as its special cases the following three cases (i), (ii) and (iii):