  In the presence of a confining potential $V$, the eigenfunctions of a
continuous Schr\"odinger operator $-\Delta +V$ decay exponentially with the
rate governed by the part of $V$ which is above the corresponding eigenvalue;
this can be quantified by a method of Agmon. Analogous localization properties
can also be established for the eigenvectors of a discrete Schr\"odinger
matrix. This note shows, perhaps surprisingly, that one can replace a discrete
Schr\"odinger matrix by \emph{any} real symmetric $Z$-matrix and still obtain
eigenvector localization estimates. In the case of a real symmetric
non-singular $M$-matrix $A$ (which is a situation that arises in several
contexts, including random matrix theory and statistical physics), the
\emph{landscape function} $u = A^{-1} 1$ plays the role of an effective
potential of localization. Starting from this potential, one can create an
Agmon-type distance function governing the exponential decay of the
eigenfunctions away from the "wells" of the potential, a typical eigenfunction
being localized to a single such well.
