We present new nonasymptotic growth and concentration bounds for a product of independent random matrices, similar in spirit to concentration for sums of independent random matrices developed in the previous decade. Our matrix product concentration bounds provide a new, direct convergence proof of Oja's algorithm for streaming Principal Component Analysis, and should be useful more broadly for analyzing the convergence of stochastic gradient descent for certain classes of nonconvex optimization problems, including neural networks. This talk covers joint work with Amelia Henriksen, De Huang, Jon Niles-Weed, and Joel Tropp.