Let S be a sample variance covariance matrix.
1. Calculate the eigenvalue decomposition of S using the function eigen().
2. Calculate the symmetric square root matrix R using the eigenvalue decomposition of S. Verify the identity S = R2.
3. Use the function chol to calculate the cholesky decomposition matrix A, and verify the identity S=AกฏA.
4. Draw a circle of center and radius 1, C(z,1), by connecting n points.
5. Transform the points that make the circle C(z,1) into an ellipse of center z and shape S E(z,S) and draw it.
6. Generate a sample X of 50 observations from a normal population with mean z and variance covariance matrix S.
7. Calculate the sample mean and sample covariance matrix of X and draw a 95% confidence ellipse region for the population mean.