1. The problem statement, all variables and given/known data Prove or give a counterexample: the product of any two selfadjoint operators on a finite-dimensional inner-product space is self-adjoint. 2. Relevant equations 3. The attempt at a solution I'd say that if we let a diagonal matrix represent T (after all, its transpose representing T*=the matrix representing T) and multiply it by a diagonal matrix representing the transformation S, then we'd end up with a diagonal matrix as a product. So the product is self adjoint since all diagonal matrices are equal to their transposes. Another case is with an nxn matrix where all entries are equal. This matrix represets T and its transpose is T*. Its matrix = its transpose so its self adjoint. Now multiplying it with another nxn matrix representing S with all entries equal to each other would obviously produce a matrix with all entries equal to each other. Or multiplying the matrix for T with a diagonal matrix would produce a diagonal matrix.