Multiplication of Eigen vectors with matrix

Question

I have vectors $c_k$ $(N \times 1)$, $k=1,\ldots$,K and a symmetric matrix $L (K \times N)$. I wish to maximize the following:

$\max_{(c_k)} \sum_{k=1}^{K} c_k^T L c_k$

Is it true to claim that the solution $c_k$ are the eigen vectors of $L$?

Thanks you!

Answer 1

Your problem is ill-formed because

• the maximum is unbounded, and

• if we constrain the vectors to be unit, in general there will be a unique $c$ that maximizes $c^TLc$, so that the solution will be $c_k=c$ for all $k$.

Also note that if the Eigenvalues are all negative, the maximum is achieved by the vector $0$.