Radon-Hurwitz Number

Question

I found this interesting segment about the Radon-Hurwitz numbers $\rho (n)$ on Wikipedia:

"For $N$ written as the product of an odd number A and a power of two $2^B$, write $$B=c+4d, 0 \le c <4.$$ Then, $$\rho(N)=2^c+8d.$$ In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real n×n matrices, for which each non-zero matrix is a similarity transformation, i.e. a product of an orthogonal matrix and a scalar matrix. " Wikipedia link

I was not able to prove this, or to find a proof online. Can someone help prove this?

Answer 1

That passage is quite confusing. $\rho(n)$ is actually the maximal possible dimension of a subspace of real $n\times n$ invertible matrices (i.e. a subspace in which every nonzero matrix is invertible). See Vector subspace of $M_n(\mathbb{R})$ with invertible matrices for some relevant literature.