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, …, 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.