Radon-Hurwitz Number 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 $$B=c+4d, 0 \le c &lt;4.$$
  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?
That passage is clearly wrong, as the union of $\{0\}$ and the set of all scalar multiples of orthogonal matrices does not constitute a real linear subspace. E.g. $\operatorname{diag}(2,0)=\operatorname{diag}(1,-1)+I_2$ is neither zero nor a scalar multiple of a real orthogonal matrix.

$\rho(n)$ 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.
That passage is clearly wrong, as the union of $\{0\}$ and the set of all scalar multiples of orthogonal matrices does not constitute a real linear subspace. E.g. $\operatorname{diag}(2,0)=\operatorname{diag}(1,-1)+I_2$ is neither zero nor a scalar multiple of a real orthogonal matrix.

$\rho(n)$ 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.