Energy of matrices

Let Mn(C) denote the space of n × n matrices with entries in C. We define the energy of A∈Mn(C) as (1)E(A)=∑k=1n|λk−tr(A)n|where λ1,…,λn are the eigenvalues of A, tr(A) is the trace of A and |z| denotes the modulus of z∈C. If A is the adjacency matrix of a graph G then E(A) is precisely the energy of the graph G introduced by Gutman in 1978. In this paper, we compare the energy E with other well-known energies defined over matrices. Then we find upper and lower bounds of E which extend well-known results for the energies of graphs and digraphs. Also, we obtain new results on energies defined over the adjacency, Laplacian and signless Laplacian matrices of digraphs.