Generalization of sum of outer product

Question

Consider a matrix $A \in \mathbb{R}^{d \times m}$ such that $m \geq d$ and denote its columns i.e $A_{:, i}$ by $a_i$. Let $AA^T$ is invertible.

Now, consider the sum $S(A) = \sum_{r=1}^m a_r a^T_r$ which is an $d \times d$ matrix.

Note that each $a_r a_r^T$ is a rank 1 matrix. Note that when m = d, it follows from another question.

Is rank(S(A)) = d ?

Example: A = \begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 1 \end{bmatrix}

S(A) = \begin{bmatrix} 2 & 1\\ 1 & 2 \end{bmatrix}

S in this example is full rank.

Answer 1

as @Omnomnomnom pointed out: Looks like $S(A) = AA^T$.