Efficient matrix multiplication

Question

I have a list of vectors vecs = {vec1, vec2, ..., vecN} where veci is a list with length $N$.

Now I have a matrix $N\times N$ called mat.

I would like to efficiently get all the numbers veci.mat.veci for $i=1$ to $N$ in a list. How do I do it?

Answer 1

This is one way:

MapThread[Dot,{vecs.mat,vecs}]

Answer 2

Here's another way:

Total[vecs.mat * vecs, {2}]

If the vectors happen to come naturally as the columns of vecs,
rather than its rows, then this will get what you want:

Total[vecs * mat.vecs]

Answer 3

One way to calculate is to leave everything in matrix form

v = RandomReal[{-1, 1}, {m=5, 10}];
mat = RandomReal[{-1, 1}, {10, 10}]; 
Diagonal[v.mat.Transpose[v]]

Since we're looking for fast ways, this seems faster:

v = RandomReal[{-1, 1}, {m=1000, 10}];
mat = RandomReal[{-1, 1}, {10, 10}];
v[[#]].mat.v[[#]] & /@ Range[m]