We dene the outer energy of a real symmetric matrix M for the eigenvalues λ 1 , …, λ n of M and their arithmetic mean λ( M ). We discuss the properties of the outer energy in contrast to the inner energy defined as E inn ( M ) = ∑ n i = 1 |λ i |. We prove that E inn is the maximum among the energy functions e : S ( n ) → R and E out among functions f ( M - λ( M )1 n ), where f is an energy function. We prove a variant of the Coulson integral formula for the outer energy.