TITLE: Tensor Notation Upper and Lower Indices
QUESTION [5 upvotes]: I want to ask what the difference between the tensors $T_i^{\; j}$ , $T_j^{\; i}$ , $T_{\; i}^{ j}$ , and $T_{\;i}^{j}$ are. In particular I am asking about the matrix representations of these tensors and their relationships.

REPLY [3 votes]: I've been wondering about this also, my conclusion from reading up in this pdf is that when using matrix representations, we are used to having the first component represent which row and the second which column. 
Therefore if we are multiplying $T^\mu _\nu v^\nu$ this can be visualised by a m,n matrix multiplying with a n dimentional column vector. 
As with the scalar product it is sometimes convenient to visualise convectors as row vectors, to maintain rules of matrix multiplication. Then one could write $w_\nu {A ^\nu} _\mu$ as the product of an n dimensional row vector with A a n,m matrix. But for example in some contexts one prefers to consider transposes multiplying with column vectors instead of using row vectors. Then one could write ${A_\mu} ^\nu w_\nu$ where one has essentially the transpose of an n,m matrix (m,n) post multiplied by a n dimentional column vector
Hence when writing these indeces, the first one from left to right represents the row and the second the column of the matrix, and they are up or down so that the latter one can be contracting with a corresponding lower or upper index of whatever comes after.
The difference between ${T^\mu} _\nu$ and ${T_\mu} ^\nu$ is that the first would be used to postmultiply by a contravariant vector $v^\nu$ and the second would be used to postmultiply by a covariant vector $v_\nu$ if one cared about maintaining representation with matrices and their products more visual. Then in this sense switching the order from left to right is equivalent to doing the transpose. 
If however one doesn't care about matrix representation and just works with the indeces then 
Disclaimer: I have never had any course on this and am just learning online, so please correct me if I misunderstood something