Derivation of the Optimal Matched Filter - Convolution vs. Correlation

On the Wikipedia page for Matched Filters here, there is a matrix algebra derivation for an optimal matched filter. Now, considering the output of an LTI system is found using the convolution operator, why is the first step of the proof on the page is to consider:

$$ y = \sum_{k=-\infty}^{\infty} h^{*}[k]x[k] $$

instead of,

$$ y[n] = \sum_{k=-\infty}^{\infty} h[n-k]x[k] $$

the convolution operation? And is $h^{*}$ the complex conjugate of the impulse response?