In the paper, the problem of diagonal stability of a pair of real matrices of a special form is studied. There is a general criterion for checking matrices for diagonal stability. However, the use of this criterion directly might be difficult. Therefore, the problem of finding classes of matrices for which it is possible to obtain constructively verifiable conditions of diagonal stability is actual. This problem has wide application in the construction of diagonal Lyapunov-Krasovskii functionals for complex systems with delay and for models of population dynamics. In the present work, constructively verifiable necessary and sufficient conditions are obtained the fulfillment of which guarantees the diagonal stability of the considered matrices.