Double vertical bars

Question

I've been reading the Wikipedia article on CSP (https://en.wikipedia.org/wiki/Common_spatial_pattern) and one of the first formulas there has double vertical bars in it. $$ {{w}}={\arg \max }_{{\mathbf {w}}}{\frac {\left\|{\mathbf {wX}}_{1}\right\|^{2}}{\left\|{\mathbf {wX}}_{2}\right\|^{2}}}$$ The text right before the formula says both the enumerator and denominator are variances of two multivariate signals. What do double vertical bars here mean? I know that double bars should mean the norm of a vector. In this case we have matrices (X1 and X2), so how can that be the norm of a matrix? Please help.

Answer 1

The norm of a matrix $A$ can be defined similarly to the norm of a vector. In your case, this $\Vert A\Vert$ means $\left(\sum_{i,j} a_{i,j}^2\right)^{1/2}$, known as the Frobenius norm. However, since $\vec w$ is likely a row-vector, the product $\vec wX$ is itself a row-vector. Thus the norm $\Vert\cdot\Vert$ is just the ordinary norm of a vector.