### Homework

The purpose of this exercise is to compute densities and distributions of quantities related to random matrices both analytically and numerically. The two quantities you would be required to compute are
• the density of the largest principal angle between random subspaces
• the distribution of the largest eigenvalue of a Wishart matrix
For both exercises you would need the mhg package here. Make sure mhg and mhgi work and compile them if necessary by typing, e.g., "mex mhgi.c" at the command prompt. Test to make sure they work. Type "mhgi(5,2,1,1,2,2)" and "hg(7,2,1.1,1.1,[1,3,3])". You should get the values 42.8667 and 656.5694, respectively.

#### Largest principal angle

Refer to the following paper. To get the largest principal angle between two p-planes in Rn, start with a random matrix A=randn(n,p). If A=QR is its QR decomposition, then the smallest singular value of the upper p-by-p submatrix of Q is the cosine our desired angle. I.e., theta = acos(min(svd(Q(1:p,1:p)))). Pick small values of n and p, e.g., n=7, p=3, and generate about 10,000 random samples of that angle. Histogram the result. Compare it with the density of the angle given analytically by Theorem 1 in the above paper. Plot both the empirical and analytical results on the same graph.

#### Largest eigenvalue of a Wishart matrix

For this exercise we need to generate a large number of Wishart matrices and compare the distribution of the largest eigenvalue with the theoretical prediction. Again, pick small values for n and p, e.g., n=3, p=4. The value of n must not exceed that of p. To generate an n-by-n random Wishart matrix A with p degrees of freedom and covariance matrix S (S is also n-by-n), one does this: B=randn(p,n)*sqrt(S); A=B'*B. Pick S to be symmetric positive definite but otherwise arbitrary. You may assume it is diagonal without any loss of generality. Generate about 10,000 Wishart matirces and get a vector of the largest eigenvalues. Compare this empirical result with the analytical prediction given, e.g., by the last equation on page 842 of this paper.