APPLIED MULTIVARIATE STATISTICS: WITH SAS® SOFTWARE

3.3. Least Squares Estimation

A natural criterion to obtain some meaningful estimators of B is to minimize with respect to the matrix B = (β₁:β₂:...:β_p). This is merely the sum of the squared deviations from the corresponding means over all observations and over all responses or dependent variables. This criterion is the same as that of minimizing tr(Y − XB)′(Y − XB), the trace (the sum of the diagonal elements) of the p by p matrix (Y − XB)′(Y − XB), resulting in the system of normal (matrix) equations

and yielding

as the least squares estimator ...