The minimum value of matrix inner product

Question

$$\begin{array}{ll} \text{minimize} & \langle L_c,W\rangle\\ \text{subject to} & \operatorname{tr}(W)=k\\ & I > W > 0\end{array}$$

in which $L_c$ is the graph Laplacian matrix

$$L_c = \mbox{diag}(C)-C, \qquad \operatorname{rank}(L_c) = n-k$$

and both $C$ and $W$ are square and their dimensions are $n$. What is the value of $W$ to get the minimum of the inner product?

Answer 1

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enter image description herek.imgur.com/PzIAY.png

which is equal to min(H'LH), and it can be resolved by [V, D] = eig(L); D = diag(D); [~, ind] = sort(D);
W = V(:,ind(1:k))*V(:,ind(1:k))'