Rankone

In linear algebra, a rank-one matrix is a matrix whose rank is one, meaning its column space has dimension one. Equivalently, it can be written as the outer product A = u v^T for some nonzero vectors u in R^m and v in R^n. If either u or v is the zero vector, A is the zero matrix with rank zero.

Consequences: all columns are multiples of a single column, and all rows are multiples of a single row; any 2x2 minor is zero. Therefore, any rank-one matrix has rank one unless it is the zero matrix.

Spectral and decomposition: A can be factorized via the singular value decomposition as A = σ x y^T with σ > 0 and unit vectors x in R^m, y in R^n; σ is the only nonzero singular value. The Frobenius norm of A equals σ. This makes rank-one matrices natural building blocks for low-rank approximations.

Rank-one updates: In numerical linear algebra, a common operation is adding a rank-one matrix, A + u v^T. If A is invertible, the Sherman–Morrison formula gives its inverse: (A + u v^T)^{-1} = A^{-1} - A^{-1} u v^T A^{-1} / (1 + v^T A^{-1} u), provided the denominator is nonzero. This enables efficient updates without full recomputation.

Applications and usage: Rank-one matrices are central to low-rank approximations, principal component analysis, data compression, and various machine learning and system identification tasks. They support storage and computational efficiency when data exhibit predominantly one-dimensional structure.

Rank-one operators: In functional analysis, a rank-one operator on a vector space with an inner product can be written as (x -> ⟨x, φ⟩ ψ), equivalently as ψ ⊗ φ. This generalizes the matrix outer product concept to infinite-dimensional settings.