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The calculation of the pseudo inverse of a matrix is intimately related to the singular value decomposition which applies to any matrix be it singular or not and square or not. The matrices involved in the singular value decomposition of a matrix A are formed with the orthogonal eigen vectors of the symmetric matrices A T A and AA T associated with their nonzero eigenvalues which forms a diagonal matrix. If instead of using the eigenvectors, which are difficult to calculate, we use any set of vectors that span the same spaces, which are easier to obtain, we can get simpler expressions for calculating the pseudoinverse, although the diagonal matrix of eigenvalues is filled. All numerical work to obtain the pseudo inverse whose components are rational numbers when the original matrix is also rational reduces to elementary row operations. We can, thus, generalize the least-squares/ minimum-length normal equations for full-rank matrices and solve said problems and obtain the pseudo inverse in terms of A and A T. without solving any eigen problems or factoring matrices.

Alternative methods of calculation of the pseudo inverse of a non full-rank matrix