---
tag: W84NTF77M0
kind: concept
name: Tensor rank
msc: [15A69, 15A03]
sources:
  - url: https://en.wikipedia.org/wiki/Tensor_rank_decomposition
---

## Summary

The rank of a tensor is the smallest number of simple tensors $v_1 \otimes \cdots \otimes v_d$ whose sum is the tensor.
For tensors of order $2$, that is matrices, it is the usual [[rank of a matrix|0ZC9YTXZRY]].
For order $3$ and higher it is much harder to understand: it can depend on the field, and computing it is NP-hard.
The rank of the matrix multiplication tensor governs the [[matrix multiplication exponent|X0G02CGQ5J]].

## Explanations

### L1 human

A simple tensor is the most basic kind of tensor, an outer product of vectors, like a rank-one matrix $u v^T$.
The rank counts how many such building blocks are needed to write a given tensor.
A matrix of rank $r$ is a sum of $r$ rank-one matrices and no fewer.
For a three-way array, the tensor $e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_2$ needs two blocks, so its rank is $2$.