So I encountered some vector/matrix equalities, such as:
let $x_i$ be column vectors of matrix $X$, then $\sum_{i} x_i x_i^T = X X^T$.
I only knew how to prove this by expanding and comparing each term in the resulting matrices on both sides, but I recently learnt a new trick: $\sum_{i} x_i x_i^T = (x_1, ... x_n)(x_1^T, ... x_n^T)^T = X X^T$ which directly finishes the proof.
Essentially this technique is treating the sum of some products as a single vector inner product. I've never seen this technique before and thought it was very useful. Are there any other general tips for simplifying vector/matrix product expressions?
https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf
"Civilization advances by extending the number of important operations which we can perform without thinking of them." - Alfred North Whitehead