In　this　paper,the　Riemannian　gradient　algorithm　and　the　natural　gradient　algorithm　are　applied　to　solve　descent　direction　problems　on　the　manifold　of　positive　definite　Hermitian　matrices,where　the　geodesic　distance　is　considered　as　the　objective　function.The　first　proposed　problem　is　the　control　for　positive　definite　Hermitian　matrix　systems　whose　outputs　only　depend　on　their　inputs.The　geodesic　distance　is　adopted　as　the　difference　of　the　output　matrix　and　the　target　matrix.The　controller　to　adjust　the　input　is　obtained　such　that　the　output　matrix　is　as　close　as　possible　to　the　target　matrix.We　show　the　trajectory　of　the　control　input　on　the　manifold　using　the　Riemannian　gradient　algorithm.The　second　application　is　to　compute　the　Karcher　mean　of　a　finite　set　of　given　Toeplitz　positive　definite　Hermitian　matrices,which　is　defined　as　the　minimizer　of　the　sum　of　geodesic　distances.To　obtain　more　efficient　iterative　algorithm　than　traditional　ones,a　natural　gradient　algorithm　is　proposed　to　compute　the　Karcher　mean.Illustrative　simulations　are　provided　to　show　the　computational　behavior　of　the　proposed　algorithms.