Notes on tensor/matrix notation

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

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

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.

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).