Rank2

Rank 2 is a term used in linear algebra to describe a matrix or linear map whose rank is two. For an m-by-n matrix A, rank(A) = 2 means the column space is a two-dimensional subspace of the ambient space (and the row space is also two-dimensional). In practical terms, there exist two columns that are linearly independent, and every column can be written as a linear combination of those two.

Equivalently, a matrix has rank 2 if it admits a rank-2 factorization A = U V^T with U in R^{m×2} and V in R^{n×2}, or if at least one 2×2 minor is nonzero while all minors of order three or larger are zero. In particular, the matrix must have at least two rows and two columns, and its row space and column space are each two-dimensional.

A simple example is the 3×3 matrix with two pivot columns, such as [[1,0,0],[0,1,0],[0,0,0]], which has rank 2. More generally, a matrix has rank exactly 2 whenever its two-dimensional subspaces capture all rows and columns.

In broader usage, rank 2 can describe a linear map whose image has dimension two, or, in tensor theory, a rank-2 tensor meaning a tensor that can be expressed as a sum of two rank-1 tensors. Rank-2 concepts are central to low-rank approximations and data compression, where a two-dimensional latent structure is extracted from higher-dimensional data.