5. The goal of this problem is to explore some properties of random matrices. Your job is to be a labora- tory scientist, performing experiments that lead to conjectures and more rened experiments. Do not try to prove anything. Do produce welldesigned plots, which are worth a thousand numbers. Dene a random mama: to be an m X m matrix Whose entries are independent samples from the real normal distribution with mean zero and standard deviation m_%. (In MATLAB, A = randn(m,m} lsqrt (111).) a) What do the eigenvalues of a random matrix look like ? What happens, say, if you take 100 random matrices and superimpose all their eigenvalues in a single plot '3 If you do this for m = 8, 16, 32, 64, . . ., what pattern is suggested? 1 2 b) The spectral radius of A is p(A) = max |A,-| where A,- are the eigenvalues of A. As in a), how does the spectral radius p(A) behave as m > oo ? c) What about norms? How does the 2norm of a random matrix behave as m ) 00 ? Why is it true that p(A) S ||A| |? Does this inequality appear to approach an equality as m > 00? d) What about condition numbers 7 or more simply, the smallest singular value Unlin? Even for xed m this question is interesting. What proportions of random matrices in Rmxm seem to have 0min 3 21,41,81, . . . ? In other words, what does the tail of the probability distribution of smallest singular value look like? How does the scale of all this change with m ? e) How do the answers (a) (d) change if we consider random triangular matrices instead of full matrices, i.e, upper-triangular matrices whose entries are samples from the same distribution as above ? \f