Given a square matrix A of order n, prove $\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$ Given $A\in F^{n \times n}$ prove:
$$\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$$
$\operatorname{rank}(A^{n+1}) \leq \operatorname{rank}(A^n)$ is easy, just from:
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
But how can I prove the other direction? or should I do it otherwise?
 A: Note that we can assume the field is algebraically closed, as the rank of the matrix does not change if we look at it as being over a larger field.
Now the matrix is similar to an upper triangular matrix. We can assume that it has a block form consisting of an upper triangular $m\times m$ matrix with only non-zero elements on the diagonal, and a block consisting of a strictly upper triangular $(n-m)\times (n-m)$ matrix. Now both the $n$'th and the $n+1$'st power of such a matrix will simply consist of some $m\times m$ upper triangular block with only non-zero elements on the diagonal (as we kill off the strictly upper triangular block when the power is at least $n-m$). This shows that these two powers have the same rank (namely $m$).
A: Coming back to this question after a few years, I've found a simpler proof, using only basic linear algebra knowledge.
First, if $\operatorname{rank}(A)=n$, use the facts:


*

*Matrix is full rank iff it is invertible

*Product of invertible matrices is invertible


so $\operatorname{rank}(A^{k})=n$ for any natural $k$.
Otherwise, use induction to show the following:

if $rank(T^k) = rank(T^{k+1})$ for some positive integer $k$, then $rank(T^k) = rank(T^m)$ for all positive integer $m \geq k$.

Finally, we have to show that if $n \gt \operatorname{rank}(A)$, then $rank(A^k) = rank(A^{k+1})$ for some $k\le n$.
$$
rank(A^k) = \dim(\operatorname{im}(A^k))
$$$$
\operatorname{im}(A) \supseteq \operatorname{im}(A^2) \supseteq \operatorname{im}(A^3) \supseteq \dots
$$
$$
n \gt \operatorname{rank}(A) \ge \operatorname{rank}(A^2) \ge \operatorname{rank}(A^3) \ge \dots \ge \operatorname{rank}(A^n) \ge \operatorname{rank}(A^{n+1}) \ge 0
$$
There are n possible values ($0,\dots,n-1$) for n+1 ranks, so there are at least two ranks that are equal.
A: Using Fitting's Lemma, one can give another version of the fine argument of @Tobias.
The sequence
$$
\ker(A) \subseteq \ker(A^2) \subseteq \ker(A^3) \subseteq \dots
$$
is ascending, and the sequence
$$
\operatorname{im}(A) \supseteq \operatorname{im}(A^2) \supseteq \operatorname{im}(A^3) \supseteq \dots
$$
is descending. Choose the smallest $m$ such that
$$
\ker(A^m) = \ker(A^{m+i}),
\qquad
\operatorname{im}(A^m) = \operatorname{im}(A^{m+i})
$$
for all $i \ge 0$. Note that if $\ker(A^m) = \ker(A^{m+1})$, then $\ker(A^m) = \ker(A^{m+i})$ for all $i \ge 0$. In particular $m \le n$.
Now Fitting's Lemma states that
$$
F^n = \ker(A^m) \oplus \operatorname{im}(A^m),
$$
and $A$ is nilpotent on the first summand, and invertible on the second one.
Then for any $k \ge m$ (actually, I believe, exactly for these values of $k$) we will have $$\operatorname{rank}(A^k) = \operatorname{rank}(A^{k+1}).$$
