Eigenvalue characterization of the capacity of discrete memoryless channels with invertible channel matrices

We investigate the capacity of finite-input finite-output discrete memoryless channels (DMC) whose channel matrix is n × n square and furthermore is assumed to be nonsingular with n linearly independent real eigenvectors. For any given DMC with such a channel matrix, we characterize the mutual information in terms of its eigenvalues. Our main result, obtained by using the method of Lagrange multipliers, is to derive an analytic expression for the capacity, depending on the eigenvectors and the eigenvalues of the invertible channel matrix. In particular, by using the inverse eigenvalue problem, we characterize the capacity of (2, 2) channels, with invertible channel matrices, in terms of lower and upper bounds that exist in the literature. In addition, numerical examples are provided, and probability of error is discussed.