  Large $N$ matrix quantum mechanics is central to holographic duality but not
solvable in the most interesting cases. We show that the spectrum and simple
expectation values in these theories can be obtained numerically via a
`bootstrap' methodology. In this approach, operator expectation values are
related by symmetries -- such as time translation and $SU(N)$ gauge invariance
-- and then bounded with certain positivity constraints. We first demonstrate
how this method efficiently solves the conventional quantum anharmonic
oscillator. We then reproduce the known solution of large $N$ single matrix
quantum mechanics. Finally, we present new results on the ground state of large
$N$ two matrix quantum mechanics.
