  An $n \times n$ matrix $A$ with real entries is said to be Schur stable if
all the eigenvalues of $A$ are inside the open unit disc. We investigate the
structure of linear maps on $M_n(\mathbb{R})$ that preserve the collection
$\mathcal{S}$ of Schur stable matrices. We prove that if $L$ is a linear map
such that $L(\mathcal{S}) \subseteq \mathcal{S}$, then $\rho(L)$ (the spectral
radius of $L$) is at most $1$ and when $L(\mathcal{S}) = \mathcal{S}$, we have
$\rho(L) = 1$. In the latter case, the map $L$ preserves the spectral radius
function and using this, we characterize such maps on both $M_n(\mathbb{R})$ as
well as on $\mathcal{S}^n$.
