Matrix algebra for the sum of products

Question

I have $T$ vectors $A_t$ $t=1,...,T$ of dimension $m\times 1$ and a vector $C$ of dimension $m \times 1$. Consider the matrix $D$ of dimension $m\times m$ defined as $$ D:=\sum_{t=1}^{T}(A_t-C)(A_t-C)' $$ Are there some algebraic properties of matrices that allow to obtain $D$ as the sum of two functions $f(\{A_t\}_{t=1,..T})$ and $g(C,T)$, i.e. $$ D=f(\{A_t\}_{t=1,..T})+g(C,T) $$?

Answer 1

Let $\widetilde A=\sum_{t=1}^T a_te_t^\top$ and $\widetilde C=C1^\top$, then $$D=(\widetilde A-\widetilde C)(\widetilde A-\widetilde C)^\top$$