# Notes on tensor/matrix notation

## Matrix multiplication of tensors

Matrix multiplication is a formal operation over 2-dimensional tables of numbers. Some of the tensors can be represented as 2-dimensional matrices:

• scalars: $$1\times 1$$

• vector contravariant and covariant components: $$n\times 1$$ and $$1\times m$$

• linear operators and bilinear forms: $$n\times m$$

Nevermind the upper and lower indices, matrix multiplication is defined by

$$(A\cdot B)_{ik}=\sum_{j}A_{ij}B_{jk}=\sum_{j}B_{jk}A_{ij}\nonumber \\$$

as you see, the order is not important once you use index notation. Whether or not something is a matrix multiplication is defined by ability to colocate summed-over index in two objects ($$ij,jk$$ above).

$$C_{ij}D_{kj}$$ is not a matrix multiplication, but because of $$D_{kj}=(D^{T})_{jk}$$, you can write $$C_{ij}D_{kj}=C_{ij}(D^{T})_{jk}=(C\cdot D^{T})_{ik}$$

You can write down in matrix notation only expressions containing components of tensors of rank at most 2.

## Upper and lower indices

By position of the index we denote whether the object is a component of a vector or a function acting on vectors (such as the basis vector).