  Consider $n$ complex random matrices $X_1,\ldots,X_n$ of size $d\times d$
sampled i.i.d. from a distribution with mean $E[X]=\mu$. While the
concentration of averages of these matrices is well-studied, the concentration
of other functions of such matrices is less clear. One function which arises in
the context of stochastic iterative algorithms, like Oja's algorithm for
Principal Component Analysis, is the normalized matrix product defined as
$\prod\limits_{i=1}^{n}\left(I + \frac{X_i}{n}\right).$ Concentration
properties of this normalized matrix product were recently studied by
\cite{HW19}. However, their result is suboptimal in terms of the dependence on
the dimension of the matrices as well as the number of samples. In this paper,
we present a stronger concentration result for such matrix products which is
optimal in $n$ and $d$ up to constant factors. Our proof is based on
considering a matrix Doob martingale, controlling the quadratic variation of
that martingale, and applying the Matrix Freedman inequality of Tropp
\cite{TroppIntro15}.
