Definition 11.1 (Canonical variates and correlations). Partition \(\underline {X} = \begin {pmatrix}\underline {X}^{(1)}\\ \underline {X}^{(2)}\end {pmatrix}\) with \(p_1\leq p_2\) and \(\Sigma = \begin {pmatrix}\Sigma _{11} & \Sigma _{12}\\ \Sigma _{21} & \Sigma _{22}\end {pmatrix}\). The first canonical correlation is \[\rho _1^{*} = \max _{\underline {a},\underline {b}} \operatorname {corr}\left (\underline {a}'\underline {X}^{(1)},\ \underline {b}'\underline {X}^{(2)}\right ),\] attained at the first pair of canonical variates \(U_1 = \underline {a}_1'\underline {X}^{(1)}\), \(V_1 = \underline {b}_1'\underline {X}^{(2)}\). Subsequent pairs maximise the correlation subject to being uncorrelated with all preceding variates.