Written exercises
Do exercises 7 and 912 of the normal distribution notes.
Numerical exercises
In this exercise you will examine the properties of the sample covariance of a normal matrix under the partial isotropy model (the “spiked covariance” model). You will simulate data from the following model:
where $\Sigma = \text{diag}( \lambda_1 + 1,\lambda_2 +1 , \lambda_3+1,1,\ldots,1)$. For each simulation scenario, you will simulate 1000 (or more) data matrices $Y$ compute the eigendecomposition of $Y^\top Y= V L V^\top$. For each simulated dataset, save the values of $v^2= \text{diag}(V)^2$, the squared diagonal elements of $V$, and the values of $L/n$.

Simulate data for the case $n=50$, $p=5$, and $(\lambda_1,\lambda_2,\lambda_3)=(9,6,3)$. Make boxplots that describe the marginal distributions of $v^2$ and the scaled eigenvalues. Repeat for the case $(\lambda_1,\lambda_2,\lambda_3)=(9,4,4)$ and compare the results. Explain any differences that you see.

Repeat 1 for the case $n=150$. Explain differences that you see.

Repeat 1 for the case $n=50$ and $p=50$ and interpret the results.

Repeat 1 for the case $n=50$ and $p=100$ and interpret the results.