  Let $\mathbf S \in \mathbb R^{n \times n}$ satisfy $\|\mathbf 1-\mathbf
S\|_2\le\epsilon n$, where $\mathbf 1$ is the all ones matrix and $\|\cdot\|_2$
is the spectral norm. It is well-known that there exists such an $\mathbf S$
with just $O(n/\epsilon^2)$ non-zero entries: we can let $\mathbf S$ be the
scaled adjacency matrix of a Ramanujan expander graph. We show that such an
$\mathbf S$ yields a $universal$ $sparsifier$ for any positive semidefinite
(PSD) matrix. In particular, for any PSD $\mathbf A \in \mathbb{R}^{n\times n}$
with entries bounded in magnitude by $1$, $\|\mathbf A - \mathbf A\circ\mathbf
S\|_2 \le \epsilon n$, where $\circ$ denotes the entrywise (Hadamard) product.
Our techniques also give universal sparsifiers for non-PSD matrices. In this
case, letting $\mathbf S$ be the scaled adjacency matrix of a Ramanujan graph
with $\tilde O(n/\epsilon^4)$ edges, we have $\|\mathbf A - \mathbf A \circ
\mathbf S \|_2 \le \epsilon \cdot \max(n,\|\mathbf A\|_1)$, where $\|\mathbf
A\|_1$ is the nuclear norm. We show that the above bounds for both PSD and
non-PSD matrices are tight up to log factors.
  Since $\mathbf A \circ \mathbf S$ can be constructed deterministically, our
result for PSD matrices derandomizes and improves upon known results for
randomized matrix sparsification, which require randomly sampling ${O}(\frac{n
\log n}{\epsilon^2})$ entries. We also leverage our results to give the first
deterministic algorithms for several problems related to singular value
approximation that run in faster than matrix multiplication time.
  Finally, if $\mathbf A \in \{-1,0,1\}^{n \times n}$ is PSD, we show that
$\mathbf{\tilde A}$ with $\|\mathbf A - \mathbf{\tilde A}\|_2 \le \epsilon n$
can be obtained by deterministically reading $\tilde O(n/\epsilon)$ entries of
$\mathbf A$. This improves the $1/\epsilon$ dependence on our result for
general PSD matrices and is near-optimal.
