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I've recently been reviewing some linear algebra, and came across the following problem:

How many similarity classes of $A \in M_{6}(\mathbb{C})$ are there such that $A$ has exactly one two-dimensional invariant subspace?

I'd like to try to understand this using the Jordan Canonical Form, or Rational Canonical Form. Here's what I've considered so far:

Thinking about eigenvalues:

Since there is exactly one 2-dimensional invariant subspace, our matrix $A$ cannot be diagonalizable, thus (since we are working over an algebraically closed field) the minimal polynomial of $A$ must contain at least one repeated root. If we have three or more distinct eigenvalues with one dimensional eigenspaces, there will be at least two 2-dimensional invariant subspaces. I conclude that the characteristic and minimal polynomials are of the forms:

$\chi(A) = (x-\alpha_{1})^{a_{1}}(x-\alpha_{2})^{a_{2}}(x-\alpha_{2})^{a_{3}}(x-\alpha_{4})^{a_{4}}$,

$m(A) = (x-\alpha_{1})^{b_{1}}(x-\alpha_{2})^{b_{2}}(x-\alpha_{2})^{b_{3}}(x-\alpha_{4})^{b_{4}}$

where $a_{1} \geq b_{1} \geq 2$, $a_{2} \geq 2$ $ a_{i} \geq b_{i} \geq 0$, $\sum_{i=1}^{i=5} a_{i} = 6$, and the $\alpha_{i}$ are distinct.

Thinking about invariant factors:

It must also be the case that a maximum of two of the invariant factors are linear polynomials, since any more would yield at least two 2-dimensional invariant subspaces in the rational canonical form. Given that the characteristic polynomial is degree 6, and (discounting multiplicity) the minimal and characteristic polynomials must have the same roots, I suppose I can count all of the possibilities satisfying the properties above. But is any list of invariant factors satisfying everything I've said actually one with exactly one 2-dimensional subspace? Or have I missed some conditions on the invariant factors implied by this property?

In general, these seem like some hairy, ad hoc considerations I'm making, and I'm hoping to find a more organized approach to this and similar problems. For example, if we generalize the problem to given $A \in M_{n}(\mathbb{C})$ with $l_{k}$ k-dimensional invariant subspaces, classify the similarity classes of such matrices, is there a good method?

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Each Jordan block contains exactly one eigenvector, this implies that there are at most two Jordan blocks. If the Jordan block is not a $1\times 1$ matrix, then it contains exactly one two dimensional invariant subspace. This two observations easily imply that the matrix should be similar to the $6\times 6$ Jordan block, i.e., there are $\mathbb C$ similarity classes (i.e., each class can be identified with the eigenvalue $\lambda\in\mathbb C$)

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  • $\begingroup$ Ah, that is much simpler than I expected, thanks! $\endgroup$ – user1348 Aug 30 '16 at 1:59

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