information:filtering, structural:summarization
Rank of a k -vector

If α ∈ ⋀ k (V) {\displaystyle \alpha \in {\textstyle \bigwedge }^{\!k}(V)}, then it is possible to express α {\displaystyle \alpha } as a linear combination of decomposable k -vectors:

where each α (i) {\displaystyle \alpha ^{(i)}} is decomposable, say

The rank of the k -vector α {\displaystyle \alpha } is the minimal number of decomposable k -vectors in such an expansion of α {\displaystyle \alpha }. This is similar to the notion of tensor rank.

Rank is particularly important in the study of 2-vectors (Sternberg 1964, §III.6) (Bryant et al. 1991). The rank of a 2-vector α {\displaystyle \alpha } can be identified with half the rank of the matrix of coefficients of α {\displaystyle \alpha } in a basis. Thus if e i {\displaystyle e_{i}} is a basis for V {\displaystyle V}, then α {\displaystyle \alpha } can be expressed uniquely as

where a i j = − a j i {\displaystyle a_{ij}=-a_{ji}} (the matrix of coefficients is skew-symmetric). The rank of the matrix a i j {\displaystyle a_{ij}} is therefore even, and is twice the rank of the form α {\displaystyle \alpha }.

In characteristic 0, the 2-vector α {\displaystyle \alpha } has rank p {\displaystyle p} if and only if