Squared canonical correlations are the eigenvalues of $\Sigma_{11}^{-1}\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}$

If $\underline X$ is a $p$-component random vector with $E(\underline X)=\underline0$ and positive definite covariance matrix $\Sigma$, partitioned into $\underline X^{(1)}$ of $p_1$ and $\underline X^{(2)}$ of $p_2$ components ($p_1+p_2=p$, $p_1\le p_2$). Then the squares of the canonical correlations are the characteristic roots of the matrix: