Did author omit trace in this problem statement?

Question

I have this problem:

Prove that the following identity is true:

$$\boldsymbol{A} \boldsymbol{A^T} = \sum_{i} \boldsymbol{a_i} \boldsymbol{a_i^T}$$

I assumed the notation $a_i$ means ith column vector of the matrix $\boldsymbol{A}$. I attempted to solve it and I believe the author of the problem meant to prove that:

$$ \text{tr}( \boldsymbol{A} \boldsymbol{A^T}) = \sum_{i} \boldsymbol{a_i} \boldsymbol{a_i^T}$$

where $\text{tr}$ is trace of a matrix. Otherwise, I get an incorrect answer.

Am I correct?

Answer 1

With the current equation ($AA^{T} = \sum\limits_{i}a_{i}a_{i}^{T}$ where $a_{i}$ is the $i$-th column of $A$), everything is fine. This is a correct equation (note that if $A$ is $m\times n$, then $a_{i}a_{i}^{T}$ is a matrix of size $m\times m$, as is $AA^{T}$, so the dimensions match up).