I'm stuck on an algebra problem. For $i=1, \dots, N$, let $Y_i \in \mathbb{R}^{r \times p}$, and $\hat{M} = \frac{1}{N} \sum_{i=1}^N Y_i$ be the matrix of the element-wise averages. Suppose $V \in \mathbb{S}_+^r$ is a positive definite $r \times r$ matrix.
I'm dealing with the sum of the quadratic terms:
$$ \frac{1}{N} \sum_{i=1}^N (Y_i - \hat{M})'V^{-1}(Y_i - \hat{M}), $$ and am curious if its possible to rewrite this sum in terms of $N-1$ summands. For example, in the case that $N = 2$, its clear that $$ \hat{M} = \frac{1}{2}Y_1 + \frac{1}{2}Y_2,$$ and consequently $$ (Y_1 - \hat{M}) = \frac{1}{2}Y_1 - \frac{1}{2}Y_2 = -(Y_2 - \hat{M}), $$ so that we can write the sum $$ \frac{1}{2} \sum_{i=1}^2 (Y_i - \hat{M})'V^{-1}(Y_i - \hat{M}) = (\frac{1}{2}Y_1 - \frac{1}{2}Y_2)'V^{-1}(\frac{1}{2}Y_1 - \frac{1}{2}Y_2).$$
Is there a generalization of this to arbitrary $N$?