0 based). 5. When implementing the algorithm, the part specified using matrix notation must be performed simultaneously. 6. This implementation does not correctly account for the case in which one dimension is an independent subspace. For example, if given a diagonal matrix, the above implementation will never terminate, as none of the eigenvalues will change. Hence, in real implementations, extra logic must be added to account for this case. Example Let Then jacobi produces the following eigenvalues and eigenvectors after 3 sweeps (19 iterations) : Applications for real symmetric matrices When the eigenvalues (and eigenvectors) of a symmetric matrix are known, the following values are easily calculated. Singular values The singular values of a (square) matrix A are the square roots of the (non-negative) eigenvalues of . In case of a symmetric matrix S we have of , hence the singular values of S are the absolute values of the eigenvalues of S2-norm and spectral radius The 2-norm of a matrix A is the norm based on the Euclidean vectornorm, i.e. the largest value when x runs through all vectors with . It is the largest singular value of A. In case of a symmetric matrix it is the largest absolute value of its eigenvectors and thus equal to its spectral radius. Condition number The condition number of a nonsingular matrix A is defined as . In case of a symmetric matrix it is the absolute value of the quotient of the largest and smallest eigenvalue. Matrices with large condition numbers can cause numerically unstable results: small perturbation can result in large errors. Hilbert matrices are the most famous ill-conditioned matrices. For example, the fourth-order Hilbert matrix has a condition of 15514, while for order 8 it is 2.7 × 108. Rank A matrix A has rank r if it has r columns that are linearly independent while the remaining columns are linearly dependent on these. Equivalently, r is the dimension of the range of A. Furthermore it is the number of nonzero singular values. In case of a symmetric matrix r is the number of nonzero eigenvalues. Unfortunately because of rounding errors numerical approximations of zero eigenvalues may not be zero (it may also happen that a numerical approximation is zero while the true value is not). Thus one can only calculate the numerical rank by making a decision which of the eigenvalues are close enough to zero. Pseudo-inverse