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On Bounds and Closed-Form Expressions for Capacities of Discrete Memoryless Channels With Invertible Positive Matrices

While capacities of discrete memoryless channels are well studied, it is still not possible to obtain a closed-form expression for the capacity of an arbitrary discrete memoryless channel (DMC). In this paper, we study a class of DMCs whose channel matrix is an invertible positive matrix. This class of channel matrices can be used to model many real-world settings. Next, an elementary technique based on Karush-Kuhn-Tucker (KKT) conditions is used to obtain (1) a good upper bound of a discrete memoryless channel having an invertible positive channel matrix and (2) a closed-form expression for the capacity if the channel matrix satisfies certain conditions related to its singular value and its Gershgorin's disk.