Is The subspace of $M_{m\times n}(\mathbb C)$ consisting of all matrices of rank equal to $k$ is connected?

Question

Let $m ,n$ be positive integers and $0 \leq k \leq \min \{m,n\}$ an integer.

Prove or disprove: The subspace of $M_{m\times n}(\mathbb C)$ consisting of all matrices of rank equal to $k$ is connected.

How should I start the problem

Answer 1

Hint : use the following result.

A matrix $M\in M_{m,n}(\mathbb{C})$ is of rank $k$ if and only if there exists $(A,B)\in GL_m(\mathbb{C})\times GL_n(\mathbb{C})$ such that :

$$AMB=\begin{pmatrix}1&0&...&...&...&...&0\\0&1&0&...&...&...&0\\\vdots&0&\ddots &0&...&...&\vdots\\ &&&1&\ddots\\0&&...&&0&...&0\\\vdots&&&&&&\vdots\\0&&...&&&...&0\end{pmatrix}$$

i.e. a matrix with $k$ ones on its diagonal and $0$ everywhere else.

Answer 2

The answer will be yes. One strategy of proof is to find a path from a rank-$k$ matrix in Jordan form to the matrix $$ \pmatrix{I_{k\times k}&0\\0&0} $$