Does this matrix operation have a name?

Let $A$ and $B$ be two $2^m$ by $2^n$ binary matrices. Then the element in the uth row and wth column of the new matrix is $ \sum_{v=0}^{2^n-1} a_{u, v \oplus w} b_{u,v}, $ where by $\oplus$ I mean XOR of the binary representations of the numbers $v$ and $w$.

So it's like to find the $uw$-th element of the new matrix I first permute the columns of $A$ in a way that depends on $w$, and then I multiply the $u$-th row of the permuted matrix by the $u$-th column of the transpose of $B$.

(This has to do with the fact that if you add two Boolean functions you find the convolution of their Walsh transforms, but I'd like to put it in the context of their correlation matrices, so it would be cool if this was actually a known matrix operation.)