Matrix $A^T A$ as sum of outer products

Question

I have recently read in a script about statistical methods in a chapter about linear regression that:

Given an $n \times k$-matrix $A$, we have

$$A^T A = \sum_{i=1}^{n} a_i^T a_i$$

where $a_i$ denotes the $i$-th row of $A$.

Unfortunately, the author doesn't give a proof of that and I can't figure out one myself. Maybe someone can help me.

source: script on page 10

Answer 1

Hint:

$$A = \left[\begin{array}{ll} a_1\\0\\ 0\\ \vdots\\0 \end{array}\right] + \left[\begin{array}{ll} 0\\a_2\\ 0\\ \vdots\\0 \end{array}\right] + \dots + \left[\begin{array}{ll} 0\\0\\ 0\\ \vdots\\a_n \end{array}\right] $$

$$ A^TA = A^T\left[\begin{array}{ll} a_1\\0\\ 0\\ \vdots\\0 \end{array}\right] + A^T\left[\begin{array}{ll} 0\\a_2\\ 0\\ \vdots\\0 \end{array}\right] + \dots + A^T\left[\begin{array}{ll} 0\\0\\ 0\\ \vdots\\a_n \end{array}\right] $$