Partial and multiple correlations and regressions: matrix calculations

Thanks to vectors and matrices (chapter 24), the partial and multiple correlations and regressions introduced in chapter 23 can still easily be used when there are more than two predictor variates. In order to broaden the discussion, let us subdivide each observed vector X into two subvectors X 1 and X 2, of which the elements are the predictor and the predicted variates respectively: $$ X = \left[ {X_1 \left| {X_2 } \right.} \right] = \left[ {X_1 \ldots X_i ,X_j \ldots X_k \left| {X_{\left( {k + 1} \right) \ldots } X_{u,} X_{v \ldots } X_q } \right.} \right]. $$