The matrix norm for an $n$-by-$n$ matrix $A$ is defined as $$|A| := \max\_{|x|=1} |Ax|$$ where the vector norm is the usual Euclidean one. This is also called the induced (matrix) norm, the operator norm, or the spectral norm. The unit ball of matrices under this norm can be considered as a subset of $\Bbb R^{n^2}$. What is the Euclidean volume of this set? 


I'd be interested in the answer even in just the $2$-by-$2$ case.



Euclidean volume of the unit ball of matrices under the matrix norm
Building on the nice answer of Guillaume: The integral


$$ \int\_{[-1,1]^n} \prod\_{i < j} \left| x\_i^2 - x\_j^2 \right| \, dx\_1 \dots dx\_n $$


has the closed-form evaluation


$$ 4^n \prod\_{k \leq n} \binom{2k}{k}^{-1}.$$


This basically follows from the evaluation of the [Selberg beta integral](http://en.wikipedia.org/wiki/Selberg_integral) Sn(1/2,1,1/2).


Combined with modding out by a typo, we now arrive at the following product formula for the volume of the unit ball of nxn matrices in the matrix norm:


$$ n! \prod\_{k\leq n} \frac{ \pi^k }{ ((k/2)! \binom{2k}{k})} .$$


In particular, we have:


* 2/3 π2 for n=2
* 8/45 π4 for n=3
* 4/1575 π8 for n=4


