Lectures 13-14: Concentration for Sums of Random Matrices

The trace of A is Tr(A) ∑di 1 Aii ∑di 1 λi(A). The trace norm of A is ‖A‖∗ ∑di 1 |λi(A)|. A symmetric matrix is positive semide nite (PSD) if all its eigenvalues are nonnegative. Note that for a PSD matrix A, we have Tr(A) ‖A‖∗. We also recall the matrix exponential eA ∑∞k 0 Ak k! which is well-de ned for all real symmetric A and is itself also a real symmetric matrix. If A is symmetric, then eA is always PSD, as the next argument shows. Every real symmetric matrix can be diagonalized, writing A UT DU, where U is an orthogonal matrix, i.e. UUT UTU I and D is diagonal. One can easily check that Ak UT DkU for any k ∈ Ž, thus Ak and A are simultaneously diagonalizable. It follows that A and eA are simultaneously diagonalizable. In particular, we have λi(eA) eλi(A). Finally, note that for symmetric matrices A and B, we have Tr(AB) 6 ‖A‖ · ‖B‖∗. To see this, let {ui} be an orthonormal basis of eigenvectors of B with Bui λi(B)ui . Then