The pseudo inverse of a matrix A is the unique matrix for which AX and XA are symmetric and for which AXA = A, XAX = X holds. If A is nonsingular, then '. When procedure jacobi (S, e, E) is called, then the relation holds where Diag(e) denotes the diagonal matrix with vector e on the diagonal. Let denote the vector where is replaced by if and by 0 if is (numerically close to) zero. Since matrix E is orthogonal, it follows that the pseudo-inverse of S is given by . Least squares solution If matrix A does not have full rank, there may not be a solution of the linear system Ax = b. However one can look for a vector x for which is minimal. The solution is . In case of a symmetric matrix S as before, one has . Matrix exponential From one finds where exp e is the vector where is replaced by . In the same way, f(S) can be calculated in an obvious way for any (analytic) function f. Linear differential equations The differential equation x' = Ax, x(0) = a has the solution x(t) = exp(t A) a. For a symmetric matrix S, it follows that . If is the expansion of a by the eigenvectors of S, then . Let be the vector space spanned by the eigenvectors of S which correspond to a negative eigenvalue and analogously for the positive eigenvalues. If then i.e. the equilibrium point 0 is attractive to x(t). If then , i.e. 0 is repulsive to x(t). and are called stable and unstable manifolds for S. If a has components in both manifolds, then one component is attracted and one component is repelled. Hence x(t) approaches as . Generalizations The Jacobi Method has been generalized to complex Hermitian matrices, general nonsymmetric real and complex matrices as well as block matrices. Since singular values of a real matrix are the square roots of the eigenvalues of the symmetric matrix it can also be used for the calculation of these values. For this case, the method is modified in such a way that S'' must not be explicitly calculated which reduces the danger of round-off errors. Note that with . The Jacobi Method is also well suited for parallelism. References Further reading Yousef Saad: "Revisiting the (block) Jacobi subspace rotation method for the symmetric eigenvalue problem", Numerical