Tensors

Tensor Ranks

The rank of a tensor is equivalent to how many [] it has

• ex. [6, 4] represents a vector and has a single pair of brackets, so is therefore of rank 1.
• recall that scalars, vectors and matrices are types of tensors

Scalar

A scalar is a tensor of rank 0, because scalars have no directional indicators, and therefore need no indices

• they do however represent quantities with magnitude

Mathematically, a scalar is of rank 0 because $N^{0} = 0$

Vector

A vector is a tensor of rank 1, meaning there is 1 index (or, basis vector) per component

• that is, when we have a vector in a 2D space, we are able to represent a vector with coordinates (e.g. $(2,4)$). Each component (here, the $x$ and the $y$ are each a component) needs only a single index.

Mathematically, a vector is of rank 1 because a vector in N-dimensional space can be represented by $N^{1} = N$ numbers.

• ex. to represent a vector in 3-Dimensional space, we need 3 indices

Matrix

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E Resources

• What's a Tensor?