14.3 Exercises

EXERCISE 14.1   Show that the eigenvalues of $\data{K}\data{K}^{\top}$ and $\data{K}^{\top}\data{K}$ are identical. (Hint: Use Theorem 2.6)

EXERCISE 14.2   Perform the canonical correlation analysis for the following subsets of variables: $\data{X}$ corresponding to $\{$price$\}$ and $\data{Y}$ corresponding to $\{$economy, easy handling$\}$ from the car marks data (Table B.7).

EXERCISE 14.3   Calculate the second canonical variables for Example 14.1. Interpret the coefficients.

EXERCISE 14.4   Use the SVD of matrix $\data{K}$ to show that the canonical variables $\eta_1$ and $\eta_2$ are not correlated.

EXERCISE 14.5   Verify that the number of nonzero eigenvalues of matrix $\data{K}$ is equal to $\mathop{\rm {rank}}(\Sigma_{XY})$.

EXERCISE 14.6   Express the singular value decomposition of matrices $\data{K}$ and $\data{K}^{\top}$ using eigenvalues and eigenvectors of matrices ${\data{K}}^{\top} \data{K}$ and $\data{K}{\data{K}}^{\top}$.

EXERCISE 14.7   What will be the result of CCA for $Y=X$?

EXERCISE 14.8   What will be the results of CCA for $Y=2X$ and for $Y=-X$?

EXERCISE 14.9   What results do you expect if you perform CCA for $X$ and $Y$ such that $\Sigma_{XY}=0$? What if $\Sigma_{XY}={\data{I}}_p$?