of . The last two properties imply the following identities: Another property is the following: if is Hermitian and idempotent (true if and only if it represents an orthogonal projection), then, for any matrix the following equation holds: This can be proven by defining matrices , , and checking that is indeed a pseudoinverse for by verifying that the defining properties of the pseudoinverse hold, when is Hermitian and idempotent. From the last property it follows that, if is Hermitian and idempotent, for any matrix Finally, if is an orthogonal projection matrix, then its pseudoinverse trivially coincides with the matrix itself, that is, . Geometric construction If we view the matrix as a linear map over the field then can be decomposed as follows. We write for the direct sum, for the orthogonal complement, for the kernel of a map, and for the image of a map. Notice that and . The restriction is then an isomorphism. This implies that on is the inverse of this isomorphism, and is zero on In other words: To find for given in , first project orthogonally onto the range of , finding a point in the range. Then form , that is, find those vectors in that sends to . This will be an affine subspace of parallel to the kernel of . The element of this subspace that has the smallest length (that is, is closest to the origin) is the answer we are looking for. It can be found by taking an arbitrary member of and projecting it orthogonally onto the orthogonal complement of the kernel of . This description is closely related to the minimum-norm solution to a linear system. Subspaces Limit relations The pseudoinverse are limits: (see Tikhonov regularization). These limits exist even if or do not exist. Continuity In contrast to ordinary matrix inversion, the process of taking pseudoinverses is not continuous: if the sequence converges to the matrix (in the maximum norm or Frobenius norm, say), then need not converge to . However, if all the matrices have the same rank as , will converge to . Derivative The derivative of a real valued pseudoinverse matrix which has constant rank at a point may be calculated in terms of the derivative of the original matrix: Examples Since for invertible matrices the pseudoinverse equals the usual inverse, only examples of non-invertible matrices are considered below. Special cases Scalars