8.6 Exercises

EXERCISE 8.1   Prove that $n^{-1} \data{Z}^{\top}\data{Z}$ is the covariance of the centered data matrix, where $\data{Z}$ is the matrix formed by the columns $z_{k}=\data{X}u_{k}$.

EXERCISE 8.2   Compute the SVD of the French food data (Table B.6).

EXERCISE 8.3   Compute $\tau_{3}, \tau_{4}, \ldots$ for the French food data (Table B.6).

EXERCISE 8.4   Apply the factorial techniques to the Swiss bank notes (Table B.2).

EXERCISE 8.5   Apply the factorial techniques to the time budget data (Table B.14).

EXERCISE 8.6   Assume that you wish to analyze $p$ independent identically distributed random variables. What is the percentage of the inertia explained by the first factor? What is the percentage of the inertia explained by the first $q$ factors?

EXERCISE 8.7   Assume that you have $p$ i.i.d. r.v.'s. What does the eigenvector, corresponding to the first factor, look like.

EXERCISE 8.8   Assume that you have two random variables, $X_1$ and $X_2=2X_1$. What do the eigenvalues and eigenvectors of their correlation matrix look like? How many eigenvalues are nonzero?

EXERCISE 8.9   What percentage of inertia is explained by the first factor in the previous exercise?

EXERCISE 8.10   How do the eigenvalues and eigenvectors in Example 8.1 change if we take the prices in $ instead of in EUR? Does it make a difference if some of the prices are in EUR and others in $?