since, in this case, . When has linearly independent rows (matrix is invertible), can be computed as This is a right inverse, as . Properties Existence and uniqueness The pseudoinverse exists and is unique: for any matrix , there is precisely one matrix , that satisfies the four properties of the definition. A matrix satisfying the first condition of the definition is known as a generalized inverse. If the matrix also satisfies the second definition, it is called a generalized reflexive inverse. Generalized inverses always exist but are not in general unique. Uniqueness is a consequence of the last two conditions. Basic properties Proofs for the properties below can be found at b:Topics in Abstract Algebra/Linear algebra. If has real entries, then so does . If is invertible, its pseudoinverse is its inverse. That is, . For any and any where (We use for "reciprocal".) More generally, for any and any the rectangular diagonal matrix where It follows that the pseudoinverse of a zero matrix is its transpose. The pseudoinverse of the pseudoinverse is the original matrix: . Pseudoinversion commutes with transposition, complex conjugation, and taking the conjugate transpose: The pseudoinverse of a scalar multiple of is the reciprocal multiple of :` for . Identities The following identity formula can be used to cancel or expand certain subexpressions involving pseudoinverses: Equivalently, substituting for gives while substituting for gives Reduction to Hermitian case The computation of the pseudoinverse is reducible to its construction in the Hermitian case. This is possible through the equivalences: as and are Hermitian. Products Suppose . Then the following are equivalent: The following are sufficient conditions for : has orthonormal columns (then ), or has orthonormal rows (then ), or has linearly independent columns (then ) and has linearly independent rows (then ), or , or . The following is a necessary condition for : The last sufficient condition yields the equalities NB: The equality does not hold in general. See the counterexample: Projectors and are orthogonal projection operators, that is, they are Hermitian (, ) and idempotent ( and ). The following hold: and is the orthogonal projector onto the range of (which equals the orthogonal complement of the kernel of ). is the orthogonal projector onto the range of (which equals the orthogonal complement of the kernel of ). is the orthogonal projector onto the kernel of . is the orthogonal projector onto the kernel