AP Goucher: Yes, Asimov's Law only applies if the matrices are orthogonally diagonalisable (<==> real symmetric). WDS: it is not correct that orthogonally diagonalisable <==> real symmetric; Any matrix M that commutes with its transpose is orthogonally diagonalisable. WF Lunnon: Now wants, given any finite set of matrices, to prove any product of n of the matrices has all eigenvalues e obeying |e|<c^n for some positive constant c. (Is that right? That is now your goal?) WDS: Well, isn't this immediately obvious by using "submultiplicative matrix norms"? The spectral norm and the Frobenius norm of the matrices are submultiplicative (so both should work), and the max |eigenvalue| is upper bounded by sqrt(FrobeniusNorm). End of proof. http://en.wikipedia.org/wiki/Matrix_norm