# Triangularizing a function matrix with smooth eigenvlaues

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.

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Did you have a look to Kato's book Perturbation Theory for Linear Operators, volume 132 of Grundlehren der mathematischen Wissenschaften ? – Denis Serre Dec 12 '12 at 7:15
@Denis Serre Yes. Kato's book discusses Jordan form. But, I find that questions about Jordan form and triangular form are a bit different. For example, the matrix $$\left(\begin{array}{cc} 1&z\\ 0&1 \end{array}\right)$$ is trivially triangulariable with smooth entries but cannot be written in Jordan form at $z=0$. – Uday Dec 12 '12 at 9:30

This is not a complete answer, but the paper

Kreiss, H. O., Über Matrizen die beschränkte Halbgruppen erzeugen, Math. Scand. 7(1959), 71-81.

contains a lot of results in this direction. Unfortunately, at the moment I do not have access to it to check...

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@András Bátkai Thank you for the reference. I will try to get this paper. Is there an English translation of this paper? – Uday Dec 12 '12 at 9:33
I am sorry to say that I cannot find an English reference... – András Bátkai Dec 23 '12 at 12:19
I have figured out a way to translate the paper. Thank you once again. – Uday Dec 27 '12 at 15:37

Not precisely what you are asking, but if you look at continuous functions (instead of smooth and homogeneous ones), Grove and Pedersen ["Diagonalizing Matrices over $C(X)$", Journal of Functional Analysis 59, 65--89, 1984] prove the following. $N \times N$ matrices can be diagonalized for all $N$ if and only if $X$ is a sub-Stonean topological space with $\dim X \leq 2$ and $X$ carries no nontrivial $G$-bundles over any closed subset, for $G$ a symmetric group or the circle group.

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