From: Henry Baker <hbaker1@pipeline.com> To: math-fun@mailman.xmission.com Sent: Thu, December 31, 2009 1:40:52 PM Subject: [math-fun] Elementary (?) matrix question I hate to be so dense about linear algebra, but here's another matrix question: Given an _arbitrary_ square matrix M with complex entries, we can compute the matrix product M.M', where M' is the conjugate transpose of M. Clearly, M.M' is Hermitian, so all of its eigenvalues are real. Question: Is there any interesting relationship between the eigenvalues of M.M' and those of M ? There's an obvious one from the fact det(M.M')=det(M)det(M'). Also, trace(M.M') is the sum of the squares of the absolute values of the entries of M. _______________________________________________ The only other evident fact is that M.M* is positive indefinite, so its eigenvalues are nonnegative. That there may not be any interesting relation between the eigenvalues of M and those of M.M* is suggested by the following example. Let M = [[1,c],[0,1]], M* = [[1,0],[c*,1]]. M has a double eigenvalue 1. M.M* = [[1 + |c|^2,c],[c*,1]] has eigenvalues b +- sqrt(b^2 - 1), b = 1 + |c|^2/2. These eigenvalues are positive and reciprocals, and can be as large (and small) as you please by choice of c. -- Gene