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Let $k$ be a field $\mathbb C$.
Consider the action of $G=GL_n(k)$ by conjugation on the set of $n\times n$ matrices over $k$.
The collection $X$ of matrices with repeated eigenvalues over $\overline k$ is a subvariety (as it is the zero set of the discriminant of the characteristic polynomial), and moreover it is preserved by $G$.
If we let $k^n\subset X$ be the diagonal matrices with repeated roots, then $Y=X\setminus G(D)$ is the set of non-diagonalizable matrices, and also has an action of $G$.

If $k=\overline k$, then every $G$-orbit contains an element in Jordan normal form, and by scaling the off-diagonal entries, we remain in the same conjugacy class, and so we see the corresponding diagonal matrix is in the closure of the orbit.
Therefore $Y$ is dense in $X$. This allows one to compute the dimension of $Y$ (I think).
However, I'm not really sure what else to say in describing $Y$.

What does $Y$ look like? I know this is a little vague, but I'm not really sure what a reasonable reformulation would be.
Are there good decompositions of $Y$ that help in understanding its structure? Is it smooth? Is it a manifold? Can we calculate useful invariants of $Y$, such as the cohomology? Are we better off understanding the individual orbits?
Are there other group actions on $Y$ which elucidate its structure?

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    $\begingroup$ Thinking about it a little, I would probably even be happy with a CW-complex structure that ignores the group action entirely, assuming there is one with a nice combinatorial description. $\endgroup$
    – Aaron
    Jul 16, 2014 at 7:10
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    $\begingroup$ You can describe each orbit: it is the quotient of $GL$ by the stabilizer of the corresponding Jordan matrix. This tells you about the shape of the union of the orbits of with fixed Jordan type (allowing the eigenvalues to change but not to change multiplities). Next: can you see how those big chunks are put together? For example: when is one such chuck in the closure of another? $\endgroup$
    – Mariano Suárez-Álvarez
    Mar 5, 2018 at 9:18
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    $\begingroup$ "If we let $k^n\subset X$ be the diagonal matrices.." I think that is $D$ instead of $k^n$ based on the definition of the quotient in the sequel. $\endgroup$
    – Joca Ramiro
    Mar 9, 2018 at 2:59
  • $\begingroup$ "every G-orbit contains an element in Jordan normal form..." Only when is concluded that $Y$ is dense in $X$ that I could infer that you were writing about a G-orbit in $Y$ in the quoted text and even then it took me some time. $\endgroup$
    – Joca Ramiro
    Mar 9, 2018 at 3:12

1 Answer 1

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The term used to describe such spaces is that of a stratified space, and there has been some progress in this sort of questions, notably for pairs of matrices.

As can be easily understood, the main question which is equivalent to understanding the stratification structure of conjugacy orbits of matrices, is the following: what Jordan forms can appear if we perturb a matrix with a given Jordan form?

This has been independently researched and understood by Boer and Thijsse in Semi-stability of sums of partial multiplicities under additive perturbation, Int. Eq. Op. Theory 3 (1980), 23–42 and Markus and Parilis in The change of the Jordan structure of a matrix under small perturbations, Mat. Issled. 54 (1980), 98–109. English translation: Linear Algebra Appl. 54 (1983), 139–152.

Using these results, one can construct the Hasse diagram of these conjugacy classes showing which classes are contained in the closure of a given one.

A nice survey about this problem is contained in: L. Klimenko, V. Sergeichuk. An informal introduction to perturbations of matrices determined up to similarity or congruence, São Paulo J. Math. Sci. 8 (2014), 1-22.

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