The Moore-Penrose pseudoinverse

Matrices that are not square have no inverse. The Moore-Penrose pseudoinverse A⁺ still lets us “solve” Ax = y as x = A⁺y. Practical algorithms compute it from the SVD, A⁺ = VD⁺Uᵀ, where D⁺ takes the reciprocal of D’s nonzero entries and then transposes the result.

What x = A⁺y gives depends on the shape of A:

• Wider than tall (more columns than rows): there can be many solutions, and A⁺y is the one with the smallest Euclidean norm ‖x‖₂.
• Taller than wide (more rows than columns): there may be no exact solution, and A⁺y is the x that makes Ax as close as possible to y, minimizing ‖Ax − y‖₂. This is the least-squares answer.