After using the transpose for a while, I wondered if there in any vague connection to other superscript stuff like exponents or something. I doubt it, but I couldn't find anything and it seems weird to have it as a superscript. Is there anything deeper going on here?
Question: Is there a reason for this notation? It's bothered me from since I learned it.
On
It's just the conventional notation and is used for the conjugate transpose too: $\mathbf A^\mathsf {H}$.
As the Wikipedia entry on the transpose of a matrix points out, you can use ${\mathbf {A}}^{\top}$ (\top) instead of $\mathbf {A}^{T}$ to make it stand out a bit better if that helps:
To avoid confusing the reader between the transpose operation and a matrix raised to the $\mathbf A^{th}$ power, the $\mathbf {A}^{\top}$ symbol denotes the transpose operation.
If $A$ is a real orthogonal matrix, then $$A^{-1} = A^T$$
This is the only little thing I can think of.