Probability of random nilpotent matrix similarity

Question

Say I pick a random $n\times n$ matrix with entries uniformly chosen from $[0, 1]$.

a) What is the probability it is nilpotent?

b) What is the probability that, given it is nilpotent, it is similar to a given matrix?

To clarify part (b), there are finitely many possibilities for it when written in Jordan normal form. What are the probabilities of each of these? (e.g. is there one of them that comes up with probability $1$?)

Edit: as Qiaochu pointed out, the probability it's nilpotent is $0$. So for part (b), can we say anything about the relative measure of the space of matrices with different Jordan blocks? For example, even if they both have measure 0, is it possible to say that the set of matrices similar to $\bigl( \begin{smallmatrix}0 & 1\\ 0 & 0\end{smallmatrix}\bigr)$ is larger in some concrete sense than those similar to $\bigl( \begin{smallmatrix}0 & 0\\ 0 & 0\end{smallmatrix}\bigr)$?

Answer 1

The nilpotent matrices form a variety of (I think) dimension $n^2-n$ in the $n \times n$ matrices (based on the fact that the coefficients of $\lambda^j$ for $j=0,\ldots,n-1$ in the characteristic polynomial must be $0$). You can then consider $n^2-n$-dimensional Hausdorff measure on this variety. Fix some nonzero vector $v$. If $A^k v \ne 0$ for $k = 1, \ldots, n-1$ then the Jordan form consists of a single block, and I'm pretty sure this will be the case for almost every nilpotent matrix.

Answer 2

Let $N$ be the set of $n\times n$ complex nilpotent matrices. $N$ is an algebraic set but it is not a pure variety because its dimension is locally constant but not globally (except for eventual singularities). Let $A\in N$ and $U$ be its Jordan form. Note that the group $Gl_n$ acts as follows:

$(P,Z)\in GL_n\times M_n\rightarrow P.Z=P^{-1}ZP\in M_n$. $A$ is in the orbit $O_U$ of $U$; $O_U$ is a variety of dimension $dim(Gl_n)-dim(S_U)$, where $S_U$ is the stabilizer $\{P;P^{-1}UP=U\}=comm(U)\cap GL_n$. Then the maximal local dimension of $N$ is reached when $dim(comm(U))$ is minimal, that is when $U$ is cyclic, that is when $x^n=0$ is its minimal polynomial. The only possible form for $U$ is $J_n$, the nilpotent Jordan block of dimension $n$ and then $dim(comm(J_n))=n$. Therefore

Proposition 1. $N$ is an algebraic set of dimension $n^2-n$ and $O_{J_n}$ is the sole component of maximal dimension.

Proposition 2. $O_{J_n}$ is a subset of $N$ that is Zariski open, then dense.

Proof. $O_{J_n}$ is characterized in $N$ by $X^{n-1}\not= 0$.

Another way of seeing things: we can go from $J_n$ to the other Jordan forms of nilpotent matrices by tending to $0$ some $1's$ of the matrix $J_n$.

Remark. The last proposition can be interpreted as follows. We randomly choose

i) a stricly upper triangular matrix $T$.

ii) a matrix $P$.

Then, "always", $P$ is invertible and $P^{-1}TP$ is a nilpotent matrix that is similar to $J_n$.