Tensor Plus

$M^i_j v^j = v^i$

We might make sense of this notation if we look at its cousin, the Matrix Notation used in Linear Algebra when we multiply two matrices (matrix is the singular) together.

$A_{mn}b_n = c_m$

Notice that n appears in both matrices.

Matrix multiplication has a rule that you go across the first matrix and down the second matrix.

Also, the count going across must equal the count going down. Said another way, n on the first matrix equals n on the second matrix.

Going across the first matrix we count columns. Going down the second matrix we count rows.

$A_{mn}B_{np}$

$A_{(rows)(columns)}B_{(rows)(columns)}$

The two notations are different, but if we’re careful we can just look at the similarities. Small letters that are subscripts or superscripts are indices that move across columns or down rows.

With this knowledge, let’s look at the first one.

$M^i_j v^j = v^i$

Only j appears in both M and v. Subscript must go across and superscripts must go down.

$M^{rows}_{column}$

Working with tensors we could end up with things with more Dimensions than what we can visualize. Rather than make up words similar to across for these extra dimensions of structure, we would prefer to just say “parse j” and it goes in the correct directions on the first and second objects.

$v^i$ is a representation. If it has three dimensions then we might represent it with the following column vector:

$\begin{bmatrix} v^1 \\ v^2 \\ v^3 \end{bmatrix}$

$v_i$ is a representation. If it has three dimensions then we might represent it with the following row vector:

$\begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix}$

The following equation contains tensors:

$A^{abjk}_{cdm} + B^{abi}_{cd} C^{ijk}_m = D^{abjk}_{cdm}$

A dummy index must appear twice in a term.

A free index must appear once in every term on both sides of the equation, and it must always be in the same location.

* * *

$\displaystyle \sum_{i=1}^2 \sum_{j=1}^3 ij$

(1)(1)+(1)(2)+(1)(3)+(2)(1)+(2)(2)+(2)(3)

$\displaystyle \sum_{j=1}^3 \sum_{i=1}^2 ij$

(1)(1)+(1)(2)+(2)(1)+(2)(2)+(3)(1)+(3)(2)

* * *

Column-stack a scalars to make $A^a$
Column-stack b $A^a$ to make $A^{ab}$ .
Column-stack j $A^{ab}$ to make $A^{abj}$ .
Column-stack k $A^{abj}$ to make $A^{abjk}$ .
Row-stack c $A^{abjk}$ to make $A^{abjk}_c$ .
Row-stack d $A^{abjk}_c$ to make $A^{abjk}_{cd}$ .
Row-stack m $A^{abjk}_{cd}$ to make $A^{abjk}_{cdm}$ .

A vector structure is built in one dimension. We need caution here–a vector expresses as many dimensions as it has components:

2 (x,y)
3 (x,y,z)
4 (ct,x,y,z)

Geometry has a problem if the structure of our object has four or more dimensions. We cannot visualize the structure of the object.

Computer science has the answer: a computer can build an array with 4 variables as easily as it builds an array with 3 variables.

This computer has no need to put a math objects into structures that we can see. It doesn’t rely on rules like go across here and go down there to tell her how to do the calculations. The problem with rules like that is different people build the structures differently and in so doing they will need different sets of rules.

If you use the wrong set of rules for a structure something really bad could happen.

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