Jacobi Eigenvalue Algorithm - Applications For Real Symmetric Matrices

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 S

2-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 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

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 .