On the energy of symmetric matrices and Coulson's integral formula

We dene the outer energy of a real symmetric matrix M for the eigenvalues λ1, …, λn of M and their arithmetic mean λ(M). We discuss the properties of the outer energy in contrast to the inner energy defined as Einn(M) = ∑ni = 1 |λi|. We prove that Einn is the maximum among the energy functions e: S(n) → R and Eout among functions f(M - λ(M)1n), where f is an energy function. We prove a variant of the Coulson integral formula for the outer energy