Wikipedia:Reference desk/Archives/Mathematics/2013 December 16

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December 16

all-invertible subspace

What is the largest dimension of a subspace of $M_{31}(\mathbb {R} )$ in which all matrices except 0 are invertible? --18:57, 16 December 2013 (UTC) — Preceding unsigned comment added by 85.65.26.40 (talk)

One dimensional. The determinant defines a homogeneous polynomial of degree 31 on the projective space of M_31. Restricting this to a line in the projective space (which is a 2-plane in M_31) gives a real polynomial of degree 31 in a real variable. Since every real polynomial of odd degree has a root, the determinant must vanish somewhere on this line. So the only linear sub spaces of the projective space in which the determinant is never zero are just the points (that is, the 1d sub spaces of M_31). Sławomir Biały (talk) 19:34, 16 December 2013 (UTC)

Interestingly, this problem seems to get much more difficult in even dimensions. In $M_{n}(\mathbb {R} )$ with $n\equiv 2{\pmod {4}}$, the identity and a complex structure on $\mathbb {R} ^{n}$ span a two-dimensional subspace containing no nonzero singular matrix. If $n\equiv 0{\pmod {4}}$ then a quaternionic structure on $\mathbb {R} ^{n}$ defines a four-dimensional subspace containing no nonzero singular matrix. More generally, when $2^{k}$ divides n, there is a faithful representation of a $2^{k}$-dimensional Clifford algebra on $\mathbb {R} ^{n}$. The degree one elements of this algebra act as invertible linear transformations (they are elements of the Pin group plus dilations). Conjecturally, this is the optimal situation, modulo details, but I don't have a proof of this. Sławomir Biały (talk) 00:17, 18 December 2013 (UTC)