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The (un)weighted stability for some matrices is one of essential hypothesesin time-frequency analysis and applied harmonic analysis. In the first part ofthis paper, we show that for a localized matrix in a Beurling algebra, itsweighted stabilities for different exponents and Muckenhoupt weights areequivalent to each other, and reciprocal of its optimal lower stability boundfor one exponent and weight is controlled by a polynomial of reciprocal of itsoptimal lower stability bound for another exponent and weights. Inverse-closedBanach subalgebras of matrices with certain off-diagonal decay can beinformally interpreted as localization preservation under inversion, which isof great importance in many mathematical and engineering fields. Let ${\mathcalB}(\ell^p_w)$ be the Banach algebra of bounded operators on the weightedsequence space $\ell^p_w$ on a simple graph. In the second part of this paper,we prove that Beurling algebras of localized matrices on a simple graph areinverse-closed in ${\mathcal B}(\ell^p_w)$ for all $1\le p<\infty$ andMuckenhoupt $A_p$-weights $w$, and the Beurling norm of the inversion of amatrix $A$ is bounded by a bivariate polynomial of the Beurling norm of thematrix $A$ and the operator norm of its inverse $A^{-1}$ in ${\mathcalB}(\ell^p_w)$.

Polynomial control on weighted stability bounds and inversion norms of localized matrices on simple graphs