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Trace has a nice geometric interpretation for a rank one operator: it is the factor by which the operator scales a vector in its image. This, together with linearity, is a geometric characterisation o …

No for $n\geq 3$. If $A\in M_n(\mathbf Z_p)$ were similar to $A^T\in M_n(\mathbf Z_p)$, then going modulo $p^2$, its image in $M_n(\mathbf Z/p^2\mathbf Z)$ would be similar to the image of its transp …

This proof is different from the one in Denis Serre's book. As usual, take $M^A$ and $M^B$ to be the $k[t]$-modules with underlying space $k^n$, where $t$ acts by $A$ and $B$ respectively. Then $A$ i …

Firstly, about "known classes" of examples. Most obviously, if $X$ itself has irreducible characteristic polynomial, in which case it does not admit invariant subspaces. A slightly more interesting e …

The problem is open, and not because nobody tried. For instance, it is known that the number of similarity classes in $M_n(\mathbf Z/p^2 \mathbf Z)$ is equal to the number of simultaneous conjugacy cl …

For Gaussian binomial coefficients we have $$ \sum_{k = 0}^n \binom nk_q = \sum_{m = 0}^\infty a_m q^m, $$ where $$ a_m = \sum_{\lambda\vdash m} \#\{k\in \mathbf Z_{\geq 0}\mid \lambda_1\leq n-k, \l …

The dimension of $V$ is the least non-negative integer $n$ such that there exist $v_1,\dotsc, v_n$ in $V$ such that there exists a unique $g\in G:=GL(V)$ that fixes each of $v_1,\dotsc,v_n$. So the is …

Put $R$ in Smith normal form. While this is usually defined for integer matrices, for a rational matrix $R$, we may write $R = P D Q$, where $P$ and $Q$ are in $GL_M(\mathbb{Z})$ and $GL_N(\mathbb{Z}) …

Treat $F^n$ as an $F[t]$-module $M^A$, where $t$ acts by the matrix $A$. Then the centralizer can be thought of as $\mathrm{End}_{F[t]} M^A$. Now, $M^A$ has a primary decomposition $ M^A = \bigoplus_ …

My answer starts off just like Emerton's answer above; you want the $G$-orbits on $G/B\times G/B$. But now, I diverge from Emerton to say that $G/B$ is the space of full flags $F_0\subset F_1\subset \ …

I address mainly Question C in the simplest special case where $R$ is $\mathbb Q$: In this case you are looking at locally nilpotent endomorphisms of a vector space. Similarity classes of such endomo …

$\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}=\begin{pmatrix}0 & 1\\0&0\end{pmatrix}$

The abelian group in question is the product of its Sylow-$p$ subgroups, which are preserved by automorphisms. Therefore the orbits in it are the products of orbits in the Sylow $p$-subgroups. Therefo …