In this paper, at the end of chapter 2, the author says that in index notation a matrix is written as $A^\mu_{\;\;\nu}$ and its transpose as $A_\nu^{\;\;\mu}$.
$A^\mu_{\;\;\nu}$ looks like a (1,1)-tensor, but in a tensor the order of indices does not matter.
- How to explain this discrepancy?
I am still trying to build intuition on tensors, so forgive me if the following questions are stupid.
- If $x$ is a vector and $A$ the matrix of a linear transformation, how to express $A^Tx$ in tensor notation?
- The linear algebra concept for a (1,1)-tensor is the matrix of a linear transformation, for a (0,2)-tensor is the matrix of a quadratic form. What it is for a (2,0)-tensor?