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Decay patterns of matrix inverses have recently attracted considerableinterest, due to their relevance in numerical analysis, and in applicationsrequiring matrix function approximations. In this paper we analyze the decaypattern of the inverse of banded matrices in the form $S=M \otimes I_n + I_n\otimes M$ where $M$ is tridiagonal, symmetric and positive definite, $I_n$ isthe identity matrix, and $\otimes$ stands for the Kronecker product. It is wellknown that the inverses of banded matrices exhibit an exponential decay patternaway from the main diagonal. However, the entries in $S^{-1}$ show anon-monotonic decay, which is not caught by classical bounds. By using analternative expression for $S^{-1}$, we derive computable upper bounds thatclosely capture the actual behavior of its entries. We also show that similarestimates can be obtained when $M$ has a larger bandwidth, or when the sum ofKronecker products involves two different matrices. Numerical experimentsillustrating the new bounds are also reported.

On the decay of the inverse of matrices that are sum of Kronecker products