What's the connection between the two “Fast Walsh Transform”?

Question

First Let's take a look at the convolution $\displaystyle C _ { i } = \sum _ { j \oplus k = i } A _ { j } * B _ { k }$, and the $\oplus$represents any boolean operation. And we are able to evaluate $C$ in $O(n \log n)$ time, using an algorithm called Fast Walsh Transform, where $n$ represents the length of the binary digits.

However, when looking at wikipedia page, it says that the Walsh Transform is to accelerate the evaluation of an $n \times n$ Matrix called Walsh Matrix. I also found it reasonable.

My question is, what's the connection between the two evaluations? I know that a convolution is a linear transformation and can be represented as a vector multiplied by a matrix. But where's the matrix in the First convolution? I am so confused, and does it represent that some multiplications with specified matrixed could be accelerated to $O(n \log n)$? (Thus the matrix is $n \times n$)