In an attempt to understand tensors I am reading this document. In page 15 we have $A^{\mu}_{\alpha}a^{\alpha}A^{\mu}_{\beta}b^{\beta}=(A^T)^{\mu}_{\alpha}A^{\mu}_{\beta}a^{\alpha}b^{\beta}$.
I can't see where that transposition comes from or why those expressions are equal. I've tried to expand those sums but I can't get the expected result (maybe I am getting confused by what those indexes in the transposed matrix mean).
So, what is $(A^T)^{\mu}_{\alpha}$ really?
It looks like there's a typo in the expression on the right side of the equation. I believe it should read $\ (A^T)^{\alpha}_{\mu}A^{\mu}_{\beta}a^{\alpha}b^{\beta}\ $.