On the pointwise matrix product and the mean value theorem

Bounds for the pointwise or Schur product of two matrices are derived with respect to the spectral norm 6·6. For real symmetric and positive semidefinite matrices Q=(q_ij) one of them gives a bound of 6|Q6, |Q|=(|q_ij|). Two of these bounds are applied to obtain a mean value theorem for g: t→f(A(t)) where A(t) is a matrix depending on a parameter t and f is a function on the spectrum of A(t).