It is also possible to define a pseudoinverse for scalars and vectors. This amounts to treating these as matrices. The pseudoinverse of a scalar is zero if is zero and the reciprocal of otherwise: Vectors The pseudoinverse of the null (all zero) vector is the transposed null vector. The pseudoinverse of a non-null vector is the conjugate transposed vector divided by its squared magnitude: Linearly independent columns If the rank of is identical to its column rank, , (for ,) there are linearly independent columns, and is invertible. In this case, an explicit formula is: It follows that is then a left inverse of : . Linearly independent rows If the rank of is identical to its row rank, , (for ,) there are linearly independent rows, and is invertible. In this case, an explicit formula is: It follows that is a right inverse of : . Orthonormal columns or rows This is a special case of either full column rank or full row rank (treated above). If has orthonormal columns () or orthonormal rows (), then: Normal matrices If is normal; that is, it commutes with its conjugate transpose, then its pseudoinverse can be computed by diagonalizing it, mapping all nonzero eigenvalues to their inverses, and mapping zero eigenvalues to zero. A corollary is that commuting with its transpose implies that it commutes with its pseudoinverse. Orthogonal projection matrices This is a special case of a normal matrix with eigenvalues 0 and 1. If is an orthogonal projection matrix, that is, and , then the pseudoinverse trivially coincides with the matrix itself: Circulant matrices For a circulant matrix , the singular value decomposition is given by the Fourier transform, that is, the singular values are the Fourier coefficients. Let be the Discrete Fourier Transform (DFT) matrix; then Construction Rank decomposition Let denote the rank of . Then can be (rank) decomposed as where and are of rank . Then . The QR method For computing the product or and their inverses explicitly is often a source of numerical rounding errors and computational cost in practice. An alternative approach using the QR decomposition of may be used instead. Consider the case when is of full column rank, so that . Then the Cholesky decomposition , where is an upper triangular matrix, may be used. Multiplication by the inverse is then done easily by solving a system with multiple right-hand sides,