Tensor rank

Tensor rank

The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix. A matrix thus has rank one if it can be written as an outer product of two nonzero vectors: $${\displaystyle A=vw^{\mathrm {T} }.}$$ The rank of a matrix ${\displaystyle A}$ is the minimum number of rank one matrices that sum to ${\displaystyle A}$: $${\displaystyle A=v_{1}w_{1}^{\mathrm {T} }+\cdots +v_{k}w_{k}^{\mathrm {T} }.}$$