By considering a C∞ structure on the ordered non-increasing of elements of Rn, we show that it is a differentiable manifold. By using of Lie groups, we show that eigenvalue function is a submersion. This fact is used to prove some results. These results is applied to prove a few facts about spectral manifolds and spectral functions. Orthogonal matrices act on the real symmetric matrices as a Lie transformation group. This fact, also, is used to prove the results.
"Lie Group Methods for Eigenvalue Function." J. Gen. Lie Theory Appl. 10 (1) 1 - 4, 2016. https://doi.org/10.4172/10.4172/1736-4337.1000240