How can one study the space of rank upto $k$ matrices of size $n\times n$ in algebraic or differential geometry over any ground field? Are "$n$ by $n$ matrices with rank $k$" an affine algebraic variety? shows this is not an affine variety.
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1Do you mean "the space of square matrices $;n\times n;$ of rank up to $;k;$ ? – DonAntonio Sep 28 '13 at 16:30
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what difference would it make if I include "upto"? – Turbo Sep 28 '13 at 16:31
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1If you don't then that set is not a (linear) space, @JAS – DonAntonio Sep 28 '13 at 16:35
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I think I mean linear. – Turbo Sep 28 '13 at 17:36
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I don't know what you mean by "study", but the space of rank $k$-matrices are defined by the vanishing of all $(k-1)\times (k-1)$ minors. – Fredrik Meyer Sep 28 '13 at 21:35
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The space $V_{n,k}$ of $n\times n$ matrices of rank $\le k$ is a very interesting mathematical object. Some of the best mathematicians in the world (working in such diverse fields as computer science, commutative algebra, algebraic geometry, linear algebra and combinatorics) worked on properties of $V_{n,k}$. Here are some references (in these papers you can find even more references):
L.B Beasley, "Null spaces of spaces of matrices of bounded rank", Current Trends in Matrix Theory (R. Grone, F. Uhlig Eds.), North-Holland, Amsterdam (1987), p. 45-50
D. Eisenbud and J. Harris, "Vector spaces of matrices of low rank", Adv. Math. 70, (1988) p. 135–155.
L. Lovasz, "Singular spaces of matrices and their application in combinatorics", Bol. Soc. Bras. Mat. 20 (1989) p. 87–99.
From what I understand, one of the key questions is to identify maximal linear subspaces in $V_{n,k}$ and their dimensions (Dieudonne problem, going back to 1930s).
Moishe Kohan
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