In　this　paper,　we　provide　two　types　of　sufficient　conditions　for　ensuring　the　quadratic　growth　conditions　of　a　class　of　constrained　convex　symmetric　and　nonsymmetric　matrix　optimization　problems　regularized　by　nonsmooth　spectral　functions.　These　sufficient　conditions　are　derived　via　the　study　of　the　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.