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I have gone through two proofs to show that matrices of rank $k$, where $0 \leq k \leq \min(m,n)$ form a submanifold of the set $ M(m \times n ,\mathbb{R}) $. Both proofs involve writing down an arbitrary matrix in the set considered, as :

$$ X = \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}$$

where $A$ is assumed to be of a minor of order $ k \times k$ that has non-zero determinant,(which I understand) and then they ask us to consider either the matrix :

$$ M_X= \begin{bmatrix} A^{-1} & -A^{-1}B \\ 0 & I_{n-k} \\ \end{bmatrix}$$

or alternatively:

$$M_X= \begin{bmatrix} I & -A^{-1}B\\ 0 & I \end{bmatrix}$$

This is followed by multiplication of $X$ with $M_X$. I understand what follows next. But this particular choice of matrix seemed to be picked out of the air as I viewed it.Could anyone please help motivate this idea??

Ref: The proof is found in J.Lee's Introduction to Smooth Manifolds, pg 117, ex 5.30.

Vishesh
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  • I apologise.I had not seen clearly that Lee does motivate the choice of $M_X$. A glaring miss. Plus a related question has already been asked. http://math.stackexchange.com/questions/518202/what-is-the-codimension-of-matrices-of-rank-r-as-a-manifold?rq=1 – Vishesh Oct 24 '13 at 09:55

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