Tensors

As a linear representation #

A tensor can be represented as a vector of x-number of dimensions. Basically, a generalization on top of scalars, vectors, and matrices. The specific “flavor” of the tensor (i.e. is it a scalar, vector, or matrix) is clarified by referring to the tensor’s “rank”. For instance; a rank 0 tensor is a scalr, rank 1 tensor is a one-dimensional vector, a rank 2 tensor is a two-dimensional vector (\(2x2\) matrix), etc.

But this is a slight simplification because a tensor, on top of being a representation of some linear descriptor, is also a geometric object.

As a geometric representation #

A more formal definition of a tensor goes as such:

an object that is invariant under a change of coordinates, and has components that change in a special, predictable way under a change of coordinates

and (more abstractly)

a collection of vectors and covectors combined together using the tensor product