  Given a symmetric matrix $A$, we show from the simple sketch $GAG^T$, where
$G$ is a Gaussian matrix with $k = O(1/\epsilon^2)$ rows, that there is a
procedure for approximating all eigenvalues of $A$ simultaneously to within
$\epsilon \|A\|_F$ additive error with large probability. Unlike the work of
(Andoni, Nguyen, SODA, 2013), we do not require that $A$ is positive
semidefinite and therefore we can recover sign information about the spectrum
as well. Our result also significantly improves upon the sketching dimension of
recent work for this problem (Needell, Swartworth, Woodruff FOCS 2022), and in
fact gives optimal sketching dimension. Our proof develops new properties of
singular values of $GA$ for a $k \times n$ Gaussian matrix $G$ and an $n \times
n$ matrix $A$ which may be of independent interest. Additionally we achieve
tight bounds in terms of matrix-vector queries. Our sketch can be computed
using $O(1/\epsilon^2)$ matrix-vector multiplies, and by improving on lower
bounds for the so-called rank estimation problem, we show that this number is
optimal even for adaptive matrix-vector queries.
