  In this paper, we provide two types of sufficient conditions for ensuring the
quadratic growth conditions of a class of constrained convex symmetric and
non-symmetric matrix optimization problems regularized by nonsmooth spectral
functions. These sufficient conditions are derived via the study of the
$\mathcal{C}^2$-cone reducibility of spectral functions and the metric
subregularity of their subdifferentials, respectively. As an application, we
demonstrate how quadratic growth conditions are used to guarantee the desirable
fast convergence rates of the augmented Lagrangian methods (ALM) for solving
convex matrix optimization problems. Numerical experiments on an
easy-to-implement ALM applied to the fastest mixing Markov chain problem are
also presented to illustrate the significance of the obtained results.
