11.2 Solution by Eigenvalues
Theorem 11.2 (Solution). The squared canonical correlations \(\rho _1^{*2}\geq \cdots \geq \rho _{p_1}^{*2}\) are the eigenvalues of \[\Sigma _{11}^{-1}\Sigma _{12}\Sigma _{22}^{-1}\Sigma _{21},\] and the vectors \(\underline {a}_i\) are its eigenvectors, with \(\underline {b}_i \propto \Sigma _{22}^{-1}\Sigma _{21}\underline {a}_i\).
Note 11.3. The pattern of this course should now be familiar. Principal components were the eigenvectors of \(\Sigma \); the discriminant of Section 7 came from the eigenvectors of \(W^{-1}B\); canonical correlations come from the eigenvectors of \(\Sigma _{11}^{-1}\Sigma _{12}\Sigma _{22}^{-1}\Sigma _{21}\). Each method poses a different question, and every one of them ends in an eigenvalue problem, which is why Section 1.9 was worth the space it took.
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