  We study the operator norm discrepancy of i.i.d. random matrices, initiating
the matrix-valued analog of a long line of work on the $\ell^{\infty}$ norm
discrepancy of i.i.d. random vectors. First, using repurposed results on vector
discrepancy and new first moment method calculations, we give upper and lower
bounds on the discrepancy of random matrices. We treat i.i.d. matrices drawn
from the Gaussian orthogonal ensemble (GOE) and low-rank Gaussian Wishart
distributions. In both cases, for what turns out to be the "critical" number of
$\Theta(n^2)$ matrices of dimension $n \times n$, we identify the discrepancy
up to constant factors. Second, we give a new analysis of the matrix hyperbolic
cosine algorithm of Zouzias (2011), a matrix version of an online vector
discrepancy algorithm of Spencer (1977) studied for average-case inputs by
Bansal and Spencer (2020), for the case of i.i.d. random matrix inputs. We both
give a general analysis and extract concrete bounds on the discrepancy achieved
by this algorithm for matrices with independent entries (including GOE
matrices) and Gaussian Wishart matrices.
