The pseudoinverse of a scalar σ is defined as 1/σ if σ ≠ 0, and zero otherwise. In the case of scalars, the pseudoinverse is the same as the inverse. You can define the pseudoinverse of a diagonal matrix by taking the scalar pseudoinverse of each entry. Then the pseudoinverse of a general real m � n matrix A, denoted by A�, is given by the following equation:
The pseudoinverse exists regardless of whether the matrix is square or rectangular, singular or nonsingular. If A is square and nonsingular, the pseudoinverse is the same as the usual matrix inverse.