TITLE: Spectral/Eigen-value solution?
QUESTION [1 upvotes]: Is there a spectral or eigen-value solution to finding $d$ vectors $x_1...x_n$ such that  
$ \sum_{i,j=1}^{d} C_{i,j} \cdot x_i^\top M x_j $ is minimized, 
with $C_{i,j}$ being a constant real-scalar and $M$ being a p.s.d real-matrix under othogonality or weighted orthogonality constraints over $x$'s? 
If not under any other constraints-what would be the solution -  to find the $x$'s?

REPLY [1 votes]: If define the matrix $\left[C\right]_{ij}=C_{ij}$ then I get that $C^\top$ is the factor. Other than that un-stated detail @dineshdileep looks exactly correct to me (though admittedly I have not done much with convex optimization). Here are my steps to bring you (close) to the first statement of his, using the same $X$ with $x$'s as columns:
\begin{align}
\sum_{i=1}^{d}\sum_{j=1}^{d}C_{i,j}x_i^\top Mx_j= & \sum_{i=1}^{d}\sum_{j=1}^{d}x_i^\top Mx_j C_{i,j} \tag{$C_{ij}$ is a scalar}\\
 = & \sum_{i=1}^{d}x_i^\top M\left(\sum_{j=1}^{d} x_j C_{i,j}\right) \tag{sum of $j$}\\
 = &  \sum_{i=1}^{d}x_i^\top M \left[XC^\top \right]_{\star i} \tag{col-mix of $X$}\\
 = & \operatorname{trace}(X^\top MXC^\top)
\end{align}
{"set_name": "stack_exchange", "score": 1, "question_id": 232619}