Wolfram Alpha:

```Index Notation (Tensors and Matrices)
-------------------------------------

Problems With Matrices
----------------------

1.  They can be big and difficult to manipulate.

2.  They do not commute so the order of multiplication
is important.

3.  Some things cannot be represented by matrices or
matrix algebra.

Consider the 2D rotation matrix:

-  -      -   -     -          - -   -
| x1 | -> | x1' | = |  cosθ sinθ || x1 |
| x2 |    | x2' |   | -sinθ cosθ || x2 |
-  -      -   -     -          - -   -

-                -
= |  cosθx1 + sinθx2 |
| -sinθx1 + cosθx2 |
-                -

Rewrite the rotation matrix as:

-         -
| Λ1'1  Λ1'2 |
| Λ2'1  Λ2'2 |
-         -

xi -> xi' = ΣΛi'jxj
j

= Λi'jxj using Einstein's notation.

= Λi'1x1 + Λi'2x2

So we get:

x1'  = cosθx1 + sinθx2

and,

x2'  = -sinθx1 + cosθx2

In index notation the order doesn't matter.  Thus,

xjΛi'j ≡ Λi'jxj

However, this not true for matrices.

-   -     -  -  -          -
| x1' | = | x1 ||  cosθ sinθ |
| x2' |   | x2 || -sinθ cosθ |
-   -     -  -  -          -

which is an illegal operation.

Converting Index to Matrix Notation
-----------------------------------

To get the matrix form we can rearrange the terms
but we need to ensure that like indeces are adjacent
to each other before contraction.  This is because
the product of 2 matrices is only defined if the
number of columns in the first matrix is equal to
the number of rows in the second matrix, i.e.

(m x n)(n x m) = m x k

Therefore,  AmnBnm will correspond to a legal matrix
operation while BnmAmn will not.

Example:

Mμν = AμσAνρMσρ:

= AνρAμσMσρ:

= AνρBμρ where Bμρ = AμσMσρ

= Aνρ(Bρμ)T

= (ABT)νμ

= ((ABT)νμ)T = (BAT)μν

But B = AM.  Therefore,

M' = AMAT```