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let $M_n(c)$ denote the n times n matrices over the complex number field. $N$ be a subspace of

$M_n(C)$.

1 If there is no unitary lies in $N$, what is the maximum of the dimension of $N$ can be?

It's easy to see that it is not less than n(n-1), I guess it's also tight, but I don't know if I am correct.

2 If all the rank of $M$ lies in $N$ are greater than a fixed integer $k$, what is the maximum of the dimension of $N$ can be?

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Very closely related question: mathoverflow.net/questions/19755. –  Scott Morrison Mar 30 '10 at 15:43
    
MathOverflow has very good $\text{\LaTeX}$ support. So there's really no excuse to have bad formatting. This looks cut-and-paste from something, what with the random line break in the first line. –  Theo Johnson-Freyd Mar 30 '10 at 21:59
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As I think I've said on a previous comment, I personally have no problem for leaving some LaTeX unformatted. I think that we should beware giving an impression that the visual presentation of a question is what's important here. –  Yemon Choi Mar 30 '10 at 22:30
    
Sorry, this is my first problem... I think these questions are interesting, and guess they might be considered before, but I don't find... –  gondolf Mar 31 '10 at 6:51
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1 Answer

In $M_n({\mathbb R})$, I studied Question 2, with $k=2$ (subspaces whose matrices are not of rank $1$) in: Formes quadratiques et calcul des variations. J. Math. Pures Appl. (9) 62 (1983), no. 2, 177--196. See a short version in: Condition de Legendre-Hadamard; espaces de matrices de rang $\not=1$. C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 1, 23--26. I don't remember whether my study was specific to the scalar field $\mathbb R$.

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It's probably easier if the field is $\mathbb{C}$. For example take $k=n$ so the question is about subspaces such that any nonzero matrix is invertible. You see that any $2$-diml space of matrices contains noninvertible matrices (because if $A,B$ are two invertible matrices, then $\det(A+X \cdot B)$ has roots in $\mathbb{C}$). However, the corresponding problem over $\mathbb{R}$ is much harder, and is related to the maximal number of linearly independent vector fields on a sphere, which involves $K$-theory. See en.wikipedia.org/wiki/Vector_fields_on_spheres for details. –  Laurent Berger Sep 21 '10 at 11:44
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