Discussion
If X be a \(p\)-component random vector with \(E(X) = 0\) and variance covariance matrix \(\Sigma\), positive definite. If X is partitioned into \(X^{(1)}\) of \(p_{1}\) components and \(X^{(2)}\) of \(p_{2}\) components such that \(p_{1} + p_{2} = p\) and \(p_{1} \le p_{2}\). Then the square of canonical correlation are the characteristic roots of the matrix :
Answer: D
The squared canonical correlations are the roots of \(\Sigma_{11}^{-1}\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}\).
Canonical correlation analysis maximises the correlation between \(a'X^{(1)}\) and \(b'X^{(2)}\). The first-order conditions lead to the eigenproblem
\[ \begin{aligned} &\Sigma_{11}^{-1}\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}\,a \\ &= \rho^{2}a \end{aligned} \]
so the squared canonical correlations are its \(p_{1}\) characteristic roots (the smaller set, since \(p_{1} \le p_{2}\)). \(\Sigma\) itself gives principal components. \(\Sigma_{11}^{-1}\Sigma_{12}\) and \(\Sigma_{22}^{-1}\Sigma_{21}\) are regression coefficient matrices, and are not even square in general.
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