The energy of graphs and matrices

Given a complex m×nm×n matrix A  , we index its singular values as σ1(A)⩾σ2(A)⩾⋯σ1(A)⩾σ2(A)⩾⋯ and call the value E(A)=σ1(A)+σ2(A)+⋯E(A)=σ1(A)+σ2(A)+⋯ the energy of A  , thereby extending the concept of graph energy, introduced by Gutman. Let 2⩽m⩽n2⩽m⩽n, A   be an m×nm×n nonnegative matrix with maximum entry α  , and ‖A‖1⩾nα‖A‖1⩾nα. Extending previous results of Koolen and Moulton for graphs, we prove thatE(A)⩽‖A‖1mn+(m−1)(‖A‖22−‖A‖12mn)⩽αn(m+m)2. Furthermore, if A is any nonconstant matrix, thenE(A)⩾σ1(A)+‖A‖22−σ12(A)σ2(A).Finally, we note that Wigner's semicircle law implies thatE(G)=(43π+o(1))n3/2 for almost all graphs G.