leftinvertibility

Leftinvertibility refers to the existence of a left inverse for a matrix or linear operator. A matrix A is left-invertible if there exists a matrix B such that B A = I, the identity on the appropriate size. In finite dimensions, this is equivalent to A having full column rank.

For a matrix A ∈ F^{m×n}, A is left-invertible precisely when rank(A) = n (so n ≤ m). In this case A has a left inverse, and the set of all left inverses is the set of matrices B ∈ F^{n×m} with B A = I_n. A common explicit left inverse is B = (A^T A)^{-1} A^T, which exists when A has full column rank (A^T A is invertible). Any left inverse satisfies B A = I_n.

Left-invertibility is contrasted with right-invertibility: a matrix A has a right inverse C with A C = I_m if and only if rank(A) = m (full row rank). If A is square, left invertibility implies right invertibility and A is invertible; in that case the left inverse and right inverse coincide with A^{-1}.

In the language of linear maps, a linear map f: V → W is left-invertible if there exists g: W → V with g ∘ f = id_V. In finite dimensions this is equivalent to f being injective, since an injective map has a left inverse, and a left inverse ensures injectivity. In infinite-dimensional settings, additional subtleties arise, and a bounded left inverse implies f is bounded below and has closed range.

Applications of left invertibility include solving linear systems, characterizing solvability, and constructing pseudoinverses in contexts where A has full column rank.