Given a matrix with function entries, which are smooth and homogeneous, and having smooth eigenvalues, can we find a conjugating matrix with smooth and homogeneous entries that triangularize the given matrix? For instance, given $A(x)$ is an $N\times N$ matrix with entries $a_{ij}(x)$ that are smooth and homogeneous in $x$ of order $1.$ Also, given that the eigenvalues of $A(x)$ are smooth. Find an invertible(may be in small neighborhood) matrix $E(x)$ with smooth entries such that $E^{-1}(x)A(x)E(x)$ is upper-triangular.

Sometime back I had asked a question on triangularizing a function matrix. Now, it is clear to me that it is possible to find, by Schur decomposition, a triangularizing matrices which are measurable. Also, one of the answers posted for that question was, it is not always possible to uniformly triangularize especially for certain matrices with non-differentiable eigenvalues. The question in this post is directed towards smoothness and homogenity of such matrices under the condition that they have smooth eigenvalues.

I would be grateful for any reference or insight in this direction. Thank you.

Perturbation Theory for Linear Operators, volume 132 of Grundlehren der mathematischen Wissenschaften ? – Denis Serre Dec 12 '12 at 7:15