rank1

Rank-1 refers to a matrix (or linear operator) whose rank is exactly one. In linear algebra, a matrix has rank 1 when its column space and row space are each one-dimensional, meaning all nonzero columns are multiples of a single column, and all nonzero rows are multiples of a single row. Equivalently, a rank-1 matrix has all its 2x2 minors equal to zero.

One common way to express a rank-1 matrix is as an outer product A = u v^T, where u and v are nonzero vectors. In this form, every column of A is a scalar multiple of u, and every row of A is a scalar multiple of v^T. Conversely, any matrix that can be written as an outer product of two vectors has rank at most 1; if u and v are nonzero, the matrix has rank exactly 1.

Key properties include: a rank-1 matrix has at most one nonzero eigenvalue, which for A = u v^T equals v^T u, and its nonzero singular value is ||u|| · ||v||. The spectrum consists of this single eigenvalue (when v^T u ≠ 0) and zeros otherwise. In terms of minors, all 2x2 minors vanish, reflecting its inability to have two independent rows or columns.

Rank-1 matrices arise naturally in low-rank approximations and updates. Any matrix A admits a best rank-1 approximation given by σ1 u1 v1^T from the singular value decomposition, where σ1 is the largest singular value and u1, v1 are corresponding singular vectors. Rank-1 updates to a matrix are used in numerical linear algebra to efficiently modify solutions without recomputing from scratch.

Example: A = [[2, 4], [1, 2]] equals u v^T with u = [2, 1]^T and v^T = [1, 2], illustrating a rank-1 structure since det(A) = 0. See also rank (linear algebra), outer product, and singular value decomposition.