applying the involution operation to the transpose of . When it does exist, it is unique. Example: Consider the field of complex numbers equipped with the identity involution (as opposed to the involution considered elsewhere in the article); do there exist matrices that fail to have pseudoinverses in this sense? Consider the matrix . Observe that while . So this matrix doesn't have a pseudoinverse in this sense. In abstract algebra, a Moore–Penrose inverse may be defined on a *-regular semigroup. This abstract definition coincides with the one in linear algebra. See also Drazin inverse Hat matrix Inverse element Linear least squares (mathematics) Pseudo-determinant Von Neumann regular ring Notes References External links Interactive program & tutorial of Moore–Penrose Pseudoinverse The Moore–Penrose Pseudoinverse. A Tutorial Review of the Theory Online Moore–Penrose Inverse calculator Matrix theory Singular value decomposition Numerical linear algebra