Suppose $\{X_k\}_{k=1}^n$ is a set of n random variables, satisfying the condition $v_{ij} = cov(X_i, X_j) \in \mathbb{R} \forall 1 \leq i,j \leq n$. Suppose all $v_{ij}$ are given. How can we find $min\{Var(\sum_{k=1}^n c_kX_k) | (c_k)_{k=1}^n \in \mathbb{R}, \sum_{k=1}^n c_k = 1\}$?
Using the properties of covariance we can transform $Var(\sum_{k=1}^n c_kX_k)$ the following way: $Var(\sum_{k=1}^n c_kX_k) = \sum_{i=1}^n \sum_{j=1}^n c_ic_jv_{ij}$.
That means that the value, we are trying to find, is the minimal value of a quadratic form, defined by matrix $(v_{ij})$, on a hyperplane, defined by the equation $\sum_{k=1}^n c_k = 1$. And here I am stuck, as I have no idea how to find it.
Any help will be appreciated.