Some results on structured \(M\)-matrices with an application to wireless communications (Q2496632)

The conditions under which a specially structured \(Z\)-matrix is an \(M\)-matrix are investigated. Given an \(n\times n\) matrix \(A\) and \(2k\) nonnegative \(n\)-vectors \(u_i,\;v_i,\;i=1,\dots,k\), three necessary and sufficient conditions are given under which the perturbation \(A-\sum_{i=1}^{k}u_iv_{i}^T\) is an \(M\)-matrix. These results are used to obtain the minimal solution of the inequality (\(A-U_kV_k^T)x\geq b\), where \(A\) is an \(n\times n\) matrix, \(U_k,\;V_k\) are nonnegative \(n\times k\) matrices with \(V_k\not= 0\), and \(b\) is a positive \(n\)-vector. This solution is employed to obtain the optimal power allocation and the characterization of the solution of the capacity problem in a CDMA communication system supporting multi-class services.