# $A$-invariant subspaces $\operatorname{Gr}(k)$

Let $$A \in GL(d,\mathbb{R})$$, which is the space of $$d\times d$$ invertible matrices. Assume that the characteristic polynomial of $$A$$ is irreducible over $$Q$$. I want to find $$A$$-invariant subspaces $$\operatorname{Gr}(k)$$, which is a Grassmannian of $$k$$-planes of $$\mathbb{R}^{d}$$.

My attempt: I think one needs to find the eigenspaces $$\wedge^{k} A$$ that are $$A$$-invariant subspaces $$\operatorname{Gr}(k)$$. If the previous sentence is correct, I have the following problems:$$A$$-invariant subspaces $$\operatorname{Gr}(k)$$ are $$k$$-dimensional, but the eigenspaces $$\wedge^{k} A$$ are $$\binom nk$$-dimensional.

• This is determined by $A$'s real canonical form, although how precisely seems tricky. WLOG, the linear operator is multiplication by $T$ on a direct sum of modules of the form $\mathbb{R}[T]/\pi(T)^n$ where $\pi$ is linear or irreducible quadratic. Invariant subspaces are not simply sums of these summands though - for instance isoclinic rotations have infinitely many invariant subspaces.
– anon
Apr 18, 2021 at 20:05
• @runway44 : Thanks for your comment. Assume that the characteristic polynomial is irreducible over $Q$. Then, can we consider the eigenspaces of the exterior products? If not, under what assumptions we can consider it?
• If the char poly is irreducible over $\Bbb Q$ then there are no repeated roots. If the char poly has no repeated roots, then (ignoring dimensions) the $A$-invariant subspaces are precisely the direct sums of the minimal ones, and there are finitely many minimal ones (you get a minimal one, in this case, by summing a conjugate pair of eigenspaces in $\Bbb C^n$ and then intersecting with $\Bbb R^n$).
• A minimal invariant subspace, as you might expect, is a proper nontrivial invariant subspace that does not contain any smaller nontrivial invariant subspaces. Extend the action of $A$ from $\Bbb R^n$ to $\Bbb C^n$, and let $V_\lambda$ be a nontrivial eigenspace for a complex number $\lambda$. Then $V_{\overline{\lambda}}$ is also a nontrivial eigenspace, where $\overline{\lambda}$ is the conjugate, and $(V_\lambda\oplus V_{\overline{\lambda}})\cap\Bbb R^n$ is a minimal invariant subspace of $\Bbb R^n$, and this is how all of them arise.