Exercise 7

Suppose that $C$ is the covariance matrix of a demeaned dataset.

1. Suppose that $\uvec$ is an eigenvector of $C$ with associated eigenvalue $\lambda$ and that $\uvec$ has unit length. Explain why $V_{\uvec} = \lambda$.

2. Suppose that the covariance matrix of a demeaned dataset can be written as $C=QDQ^T$ where $$Q = \begin{bmatrix} \uvec_1 & \uvec_2 \end{bmatrix}, \hspace{24pt} D = \begin{bmatrix} 10 & 0 \\ 0 & 0 \\ \end{bmatrix}.$$ What is $V_{\uvec_2}$? What does this tell you about the demeaned data?

3. Explain why the total variance of a dataset equals the sum of the eigenvalues of the covariance matrix.