  The paper develops a technique for solving a linear equation $Ax=b$ with a
square and nonsingular matrix $A$, using a decentralized gradient algorithm. In
the language of control theory, there are $n$ agents, each storing at time $t$
an $n$-vector, call it $x_i(t)$, and a graphical structure associating with
each agent a vertex of a fixed, undirected and connected but otherwise
arbitrary graph $\mathcal G$ with vertex set and edge set $\mathcal V$ and
$\mathcal E$ respectively. We provide differential equation update laws for the
$x_i$ with the property that each $x_i$ converges to the solution of the linear
equation exponentially fast. The equation for $x_i$ includes additive terms
weighting those $x_j$ for which vertices in $\mathcal G$ corresponding to the
$i$-th and $j$-th agents are adjacent. The results are extended to the case
where $A$ is not square but has full row rank, and bounds are given on the
convergence rate.
