Is this a theorem? Let A and B be real symmetric matrices all of whose eigenvalues are positive. Then all eigenvalues of AB are positive. -- Gene ______________________________________________ A few remarks: 1. The product AB need not be symmetric. For nonsymmetric matrices, positive eigenvalues is stronger than positive-definite. The matrix [[1,1],[-1,1]] is positive-definite, but has eigenvalues 1 +- i. I'm looking for the stronger conclusion of positive eigenvalues. 2. If A and B are only required to be positive-definite, and not necessarily symmetric, then AB need not be positive-definite. For a counterexample, let A = B be a rotation in the plane by 60 degrees. 3. If A and B are only required to be real symmetric, then AB need not have real eigenvalues. Example: A = [[1,0],[0,-1]], B = [[0,1],[1,0]], AB = [[0,1],[-1,0]]. -- Gene