  We show that the sparsified block elimination algorithm for solving
undirected Laplacian linear systems from [Kyng-Lee-Peng-Sachdeva-Spielman
STOC'16] directly works for directed Laplacians. Given access to a
sparsification algorithm that, on graphs with $n$ vertices and $m$ edges, takes
time $\mathcal{T}_{\rm S}(m)$ to output a sparsifier with $\mathcal{N}_{\rm
S}(n)$ edges, our algorithm solves a directed Eulerian system on $n$ vertices
and $m$ edges to $\epsilon$ relative accuracy in time $$ O(\mathcal{T}_{\rm
S}(m) + {\mathcal{N}_{\rm S}(n)\log {n}\log(n/\epsilon)}) +
\tilde{O}(\mathcal{T}_{\rm S}(\mathcal{N}_{\rm S}(n)) \log n), $$ where the
$\tilde{O}(\cdot)$ notation hides $\log\log(n)$ factors. By previous results,
this implies improved runtimes for linear systems in strongly connected
directed graphs, PageRank matrices, and asymmetric M-matrices. When combined
with slower constructions of smaller Eulerian sparsifiers based on short cycle
decompositions, it also gives a solver that runs in $O(n \log^{5}n \log(n /
\epsilon))$ time after $O(n^2 \log^{O(1)} n)$ pre-processing. At the core of
our analyses are constructions of augmented matrices whose Schur complements
encode error matrices.
