  Decay patterns of matrix inverses have recently attracted considerable
interest, due to their relevance in numerical analysis, and in applications
requiring matrix function approximations. In this paper we analyze the decay
pattern 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, $I_n$ is
the identity matrix, and $\otimes$ stands for the Kronecker product. It is well
known that the inverses of banded matrices exhibit an exponential decay pattern
away from the main diagonal. However, the entries in $S^{-1}$ show a
non-monotonic decay, which is not caught by classical bounds. By using an
alternative expression for $S^{-1}$, we derive computable upper bounds that
closely capture the actual behavior of its entries. We also show that similar
estimates can be obtained when $M$ has a larger bandwidth, or when the sum of
Kronecker products involves two different matrices. Numerical experiments
illustrating the new bounds are also reported.
