Is there a way to reduce the covariance matrix of several source signals to the dominant source signal?
What you want to do is dimension reduction. The most basic, yet very powerful and commonly used, technique to do it is principal component analysis (PCA). PCA operates on the covariance matrix. You can look it up on Wikipedia for a throughout tutorial.

PCA decomposes your data $y$, having dimension $d$, into $d$ principal components. The first principal component explains the most variance in your data, the next component the second most variance, and so on. Throw off the last $k$ principal components $-$ and ta-daa $-$ you reduce dimension. To answer your question, take $k = d - 1$ and then you get the most dominant component (in terms of explained variance).

(Tip: MATLAB and similar software usually come with an easy-to-use function to perform PCA).