The volume of the unit ball for the spectral norm in nxn real matrices is given by the formula
$$ c_n \int\limits_{[-1,1]^n} \prod_{i < j} |x_i^2-x_j^2| dx_1\dots dx_n $$
where $c_n = n! 4^{-n} \prod_{k=1}^n v_k^2$
and $v_k=\pi^{k/2}/\Gamma(1+k/2)$ is the volume of the unit ball in R^n.
A much more general formula for calculating all kind of similar quantities appears e.g. here (Lemma 1). The proof is by applying the SVD decomposition as a change of variables.
The first values are
- 2/3 π2 for 2x2 matrices
- 8/45 π4 for 3x3 matrices
- 4/1575 π8 for 4x4 matrices ...
There might be a closed formula for the integral above. Edit : such a formula now appears in Armin's answer!!
