Math behind Multi-class linear discriminate analysis (LDA)

I have a question about Linear Discriminant Analysis (LDA) for the purpose of Dimensionality Reduction.

So I understand for the algorithm to calculate for $$k$$ projection vector(s) you need to determine the eigenvector(s) that corresponds to the top $$k$$ eigenvalue(s). But can anyone explain what you do with those eigenvectors after you have calculated them to get a final output?

My guess is to multiply all of the eigenvectors (projection vectors) together and then multiply that with each point, $$x$$, in the original dataset to produce a new corresponding point $$y$$. Does this sound right?

Given normalized top $$k$$ eigenvectors $$v_1,\ldots,v_k$$, you send a point $$x$$ to the tuple $$(\langle x,v_1 \rangle, \ldots, \langle x,v_k \rangle)$$.
Alternatively, you put the eigenvectors as rows in a matrix $$M$$, and you map the column vector $$x$$ to $$Mx$$ (this is exactly the same thing as above).
• Right, $y = Mx$ and so $y_i = (Mx)_i$. Commented May 3, 2020 at 17:57