# ~ Index notation

Index notation is a way of writing tensors using indices.

## Einstein summation convention

The Einstein summation convention is a way of simplifying the notation of summation over indices that range over a set. For example, given indices that range over the set ${1,2,3}$, we write

$$y = \sum_{i = 1}^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3,$$

which is simplified by the Einstein notation as such:

$$y = c_i x^i.$$

### Identities

1. $a_{ij}(x_j + y_j) \equiv a_{ij}x_j + a_{ij} y_y$
2. $a_{ij} x_i y_j = a_{ij} y_j x_i$
3. $a_{ij} x_i x_j = a_{ji} x_i x_j$
4. $(a_{ij} + a_{ji})x_i x_j = 2a_{ij} x_i x_j$
5. $(a_{ij} - a_{ji})x_i x_j = 0$

### Non-identities

1. $a_{ij}(x_i + y_j) \ne a_{ij}x_i + a_{ij} y_j$
2. $a_{ij} x_i y_j \ne a_{ij} x_j y_i$
3. $(a_{ij} + a_{ji})x_i y_j \ne 2a_{ij} x_i y_j$

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