Exercise 1

Suppose that $$Q = \begin{bmatrix} -1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} \\ \end{bmatrix}, \hspace{24pt} D_1 = \begin{bmatrix} 75 & 0 \\ 0 & 74 \end{bmatrix}, \hspace{24pt} D_2 = \begin{bmatrix} 100 & 0 \\ 0 & 1 \end{bmatrix}$$ and that we have two datasets, one whose covariance matrix is $C_1 = QD_1Q^T$ and one whose covariance matrix is $C_2 = QD_2Q^T$. For each dataset, find

1. the total variance.

2. the fraction of variance represented by the first principal component.

3. a verbal description of how the demeaned data points appear when plotted in the plane.