Spectral/Eigen-value solution? 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?
 A: I am not sure if this helps, but it will be good if your $C$ have some nice properties. Define the matrix $X=[x_1,x_2,...,x_n]$ with the vectors under question as columns. Then you can rewrite your problem as 
\begin{align}
\min_{X}~trace(CX^{T}MX)
\end{align}
Define $M=F^{T}F$ ($M$ being a psd matrix, you will be able to find such a $F$). Define $Y=FX$. You get the above problem as 
\begin{align}
\min_{Y}~trace(CY^{T}Y)
\end{align}
Now $trace(CY^{T}Y)=trace(YCY^{T})$, (circularity of trace). At this point, to progress any further, we need to have some properties on $C$. For instance, if $C$ is symmetric, you can decompose as you did with $M$ and come up with a even simpler condition. 
EDIT---
Starting from the last point, define $Z=Y^{T}Y$. Then you can rewrite the problem as
\begin{align}
\min_{Z}~trace(CZ) \\
subject ~to~Z\geq0
\end{align}
where $Z\geq 0$ means it should be psd. This is a convex optimization problem with a linear objective and a semi-definite constraint. It is referred to as semi-definite programming. If you do not have any nice properties on $C$, I think this is what you can do. 
A: 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}
