Why does the space of real $N \times N$ rank-$k$ matrices form a manifold?

Question

In this paper, Uwe Helmke and Mark Shayman claim that it follows because the rank-$k$ matrices with signature $I_{p} - I_{q}$ are an orbit of a group action of $\mbox{GL}(N)$, but I am not sure what fact is being referenced here.

Answer 1

Notice that the set of all $n\times k$ matrices with rank $k$ is an open set of $\mathbb R^{nk}$, as a matrix of this size has rank $k$ iff one its minors of size $k\times k$ is nonzero.

If we take an $n\times n$ matrix $M$ of rank $k$ whose first $k$ columns are linearly independent, the last $n-k$ columns are linear combinations of the first $k$ columns, and such a combination can be codified uniquely using the ordered basis $(M^1,\ldots,M^k)$ and an element of $\mathbb R^k$. Thus the set of all square matrices of size $n$ of rank $k$ whose first $k$ columns are linearly independent is diffeomorphic to $U\times \mathbb R^{k(n-k)}$ for some open subset $U$ of $\mathbb R^{kn}$.

Modifying this construction, you can build an atlas of the set of all square matrices of size $n$ of rank $k$ consisting of $n\choose k$ maps, whose changes of coordinates are all $C^\infty$.

Therefore this set is a manifold of dimension $2nk-k^2$.

Answer 2

Since you're asking about signature and $N \times N$ matrices, I assume you mean to restrict to symmetric matrices, as the authors do in Section 2 of the linked paper. (As Camilo's answer shows, this is not a necessary restriction, and indeed the authors treat the case of general $M \times N$ matrices in Section 4, but there they do not discuss signature.)

This is an application of Sylvester's Law of Inertia, which says that for any real-quadratic form $Q$ on a finite-dimensional real vector space $\Bbb V$ there is a basis with respect to which the matrix representation $[Q]$ of $Q$ is $$I_p \oplus -I_q \oplus 0_r ,$$ and that the constants $p, q, r$ are independent of the choice of such a basis. By inspection, $\operatorname{rank} Q = p + q$.

Concretely, under a change of basis encoded by a matrix $P \in GL(n, \Bbb R)$, the matrix representation of $[Q]$ transforms via congruence, i.e., as $[Q] \rightsquigarrow P^\top [Q] P$. This defines a (smooth) action of $GL(N, \Bbb R)$ on the space $S(N, \Bbb R)$ of symmetric $N \times N$ matrices, and in this language Sylvester's Law of Inertia says that the orbits of this action are precisely the sets of matrices with a given signature $(p, q)$. One can check that stabilizer $H_{p, q}$ in $GL(N, \Bbb R)$ of $I_p \oplus -I_q \oplus 0_r$ is closed (it's the solution set to a particular polynomial system), so the symmetric matrices of signature $(p, q)$ comprise a smooth homogeneous manifold $M_{p, q}$ that we can identify with $GL(N, \Bbb R) / H_{p, q}$.