This might be a very stupid question, but I do not seem to understand why I would multiple a matrix with its transpose. I am not a mathematician, but I am very interested in understanding the practical usage of equations:
Imagine I have three products sales Apple, Orange and Pear for the last 3 days in a matrix form called A: $$ A= \begin{bmatrix} Apple & Orange & Pear \\ 10 & 2 & 5 \\ 5 & 3 & 10 \\ 4 & 3 & 2 \\ 5 & 10 & 5 \\ \end{bmatrix}$$
What will $AA^{\rm T}$ tell me?
I have seen this long answer link: Is a matrix multiplied with its transpose something special?, but I did not get it at all.
I see that a lot of equations use the product $AA^{\rm T}$ and I really hope that someone will give a very simple answer.
Lets consider the matrix $A$ characterizing the values of some variables $a_{ij}$, $j=1...m$ with values at different times $i=1...n$, as in the OP example, but transposed.
If the variables are normalized in mean, the matrix $\frac 1m A^TA$ is the estimator of the covariances $s_{j_1j_2}=\mathbb{E}(a_{\cdot j_1}a_{\cdot j_2}) \approx \frac 1m \sum a_{j_1}a_{j_2}$ for the set of random variables $a_{\cdot j=1...m}$.
If the entries $a_{ij}$ of $A$ have units of $[a]$, then the entries of $AA^T$ will have units of $[a^2]$. This is consistent with the abovementioned.
When solving the problem $Ax=B$, the solution $x=(A^TA)^{-1}A^TB$ is the best estimator (LS), provided that the covariance as defined above, is enough variable to be invertible.