What is the difference in writing $a^{i}_j$ and $a_{i}^j$ for a matrix?

I'm trying to self study tensor calculus. I was trying to derive the notation for covariant and contravariant indexes of a linear transformation matrix ($(1,1)$ type tensor).

So I did the following: Try for the $2 \times 2$ case, and then try to find a pattern.

For covectors (covariant) $x_i$: (I shall first assume that both indexes of the matrix are up as contravariant, just for simplicity of notation. I'll later "correct" this according to what i have found).

We have:

$$\begin{bmatrix} x_1' & x_2 ' \end{bmatrix} = \begin{bmatrix} x_1 & x_1 \end{bmatrix} \begin{bmatrix} a^{11} & a^{12} \\ a^{21} & a^{22} \end{bmatrix}$$ And so, for $x_j'$, I'll have: $$x_j' = a^{ij}x_i$$ (Summation convention here). Here, I am summing on the first index of the matrix. For the contravariant case, I have: $$x^i ‘= a^{ij}v^j$$ Hence summing on the second index of the matrix. So, I tought about writing $a$ as $a_{i}^j$, calling then $i$ as the covariant index and $j$ as the contravariant index. Does this make any sense? The first problem I see is that this goes against the summation convention, who states that the indexes must be summed up when they are at different positions (ex: $a^iv_i$ would mean a summation in $i$, but $a_ iv_i$ would not).

This construction I've made would be equivalent to: $$a = a_ib^j \mathbf{e}_i\otimes \mathbf{e}^j$$

My other question is if it is possible arriving to the following:

$$a = a^ib_j \mathbf{e}^i\otimes \mathbf{e}_j$$

Using matrix algebra? Is that possible?

I'm really confused. I've seen 2nd order tensors being written as $a^i_j$ and as $a_i^j$. What is the difference between them? Do they act the same way on vectors?

• take a look at the math.stackexchange.com/questions/1047994/… where is used a std convention on indexation of rank two tensors – janmarqz Jun 3 '18 at 21:10
• @janmarqz Thank you. However, while the answer clarify things to me a little bit, I am not yet “satisfied” with the explanation. I would like to see the possibilities of relation between a tensor construction and matrix algebra, if that is even possible, and the meaning behind the indexes placement. – Vitor C Goergen Jun 4 '18 at 10:40
• The things for rank two tensors what you see there say that for the 4 kinds of these tensor the indexation are suited to keep information organized, that is a math viewpoint which is pretty employed into math branches like geometry and analysis, but for an outside math meaning you need to look at its classical applications which are in the physical sciences like electromagnetism, relativity and even quantum mechanics among many more – janmarqz Jun 4 '18 at 16:09
• I’ll ask the same question in PhysicsSE. – Vitor C Goergen Jun 5 '18 at 22:08
• ok, keep asking and don't forget to upvote :) – janmarqz Jun 6 '18 at 0:21