Some Aspects of the Solution of Singular Normal Equations with the Use of Linear Restrictions
Arthur A. Rayner, R. M. Pringle · SIAM Journal on Applied Mathematics · 1976
A method of solving normal equations $X'X{\bf b} = X'{\bf y}$ in the linear model not of full rank is to apply a convenient set of linear restrictions $C{\bf b} = {\bf c}$, where C is complementary to $X'\,X$ and ${\bf c}$ is usually ${\bf 0}$ A characterization is derived of all generalized inverses of $X'X$ which produce the solution subject to $C{\bf b} = {\bf 0}$. This leads to the finding that the $g_3 $-inverse of X subject to $CX^{g_3 } = 0$ is unique and to the complementary result that $C^{g_3 } $ subject to $XC^{g_3 } = 0$ is unique. A case is made on this basis for discarding Plackett’s matrix $D[ 7 ]$, which appears in some books. A general formula is derived for obtaining the solution under one set of restrictions from that under a different set. The set of left inverses of $[ {\begin{array}{*{20}c} X \\ C \\ \end{array} } ]$, where C has full row-rank, is considered.