| # Pseudo-Inverse of a Matrix The pseudo-inverse of a matrix is a matrix that generalizes to arbitrary matrices the notion of inverse of a square, invertible matrix. The pseudo-inverse can be expressed from the singular value decomposition (SVD) of , as follows. Let the SVD of be where are both orthogonal matrices, and is a diagonal matrix containing the (positive) singular values of on its diagonal. Then the pseudo-inverse of is the matrix defined as Note that has the same dimension as the transpose of . This matrix has many useful properties: If is full column rank, meaning , that is, is not singular, then A is a left inverse of , in the sense that . We have the closed-form expression If is full row rank, meaning , that is, is not singular, then A is a right inverse of , in the sense that . We have the closed-form expression If is square, invertible, then its inverse is . The solution to the least-squares problem with minimum norm is . Example: pseudo-inverse of a matrix.