Hi Everybody!

Given a matrix, with smooth functions as arguments is there any result which say about its triangularization?

I know that, the question is in affirmative for diagonalizing a matrix which has distinct eigenvalues. Infact, we can show by Implicit function theorem that eigenvalues can be chosen to be $\mathcal{C}^{1}$functions. We can also calculate projection operators for the matrix using Cauchy integral formula. 

But, I have not found any result for triangularization. An answer or any references would be great help. Thank you.