Jacobi eigenvectors

Note that since SVD is based on eigenvector decompositions of cross-product matrices, this algorithm gives equivalent results as the Jacobi rotation when the sample covariance matrix C is used. This means that SVD will not allow a robust PCA solution however, for Jacobi rotation a robust estimation of the covariance matrix can be used. 

Calculation of eigenvectors requires an iterative procedure. The traditional method for the calculation of eigenvectors is Jacobi rotation (Section 3.6.2). Another method—easy to program—is the NIPALS algorithm (Section 3.6.4). In most software products, singular value decomposition (SVD), see Sections A.2.7 and 3.6.3, is applied. The example in Figure A.2.7 can be performed in R as follows ... 

The eigenanalysis of the MIL tensor is run via Jacobi method to calculate the main characteristics values, that is, eigenvalues (eo-g), and characteristic directions, that is, eigenvectors (co-s)-... 

This is the orthogonality relation of the two Lanczos polynomials Q (m) and Qm(u) with the weight function, which is the residue dk [48]. We recall that the sequence Q = (Q ( z-) coincides with the set of eigenvectors of the Jacobi matrix (60). 

This procedure transforms the ill-conditioned problem of finding the roots of a polynomial into the well-conditioned problem of finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix. As shown by Wilf (1962), the N weights can then be calculated as Wa = OToV ai where tpai is the first component of the ath eigenvector (pa of the Jacobi matrix. 

Below a Matlab script for the calculation of a quadrature approximation of order N from a known set of moments iti using the Wheeler algorithm is reported. The script computes the intermediate coefficients sigma and the jacobi matrix, and, as for the PD algorithm, determines the nodes and weights of the quadrature approximation from the eigenvalues and eigenvectors of the matrix. 

This is a straightforward implementation of the old faithful Jacobi method. The eigenvalues and eigenvectors are generated in order of lowest eigenvalue first. 

The method which has been implemented to generate the eigenvalues and eigenvectors of a real symmetric matrix is not, in fact, the fastest method there are methods which have asymptotic dependence on floating point operations, while the Jacobi method depends asymptotically on m. f77 implementations of these methods (the Givens and Householder methods) are available for most computers and calls to eigen may be simply replaced by corresponding calls to the other routine. 

Consider first the matrix-vector product for a triatomic sequential diagonalization-truncation basis of the type discussed in Section 4. In this section we label with j, functions of q and q2 obtained by diagonalizing a two-dimensional Hamiltonian for each DVR point (<73 )y for coordinate q. If the two-dimensional Hamiltonian is diagonalized in a direct product q q2 basis and a and are DVR labels for the q and q2 DVR basis functions, then the matrix of eigenvectors is the transformation matrix whose elements are (Instead of diagonalizing the two-dimensional Hamiltonian in a direct product DVR basis one might use products of optimized ID functions for q and DVR functions for q2 ) In a basis of functions labeled by j and y the Hamiltonian (written in Radau, symmetrized Radau, or Jacobi coordinates) matrix elements are,... 

As a demonstration of the useMness of Gershgorin s theorem, we generate a convergence criterion for the Jacobi iterative method of solving Ax = b. This example is typical of the use of eigenvalues in numerical analysis, and also shows why the questions of eigenvector basis set existence raised in the next section are of such importance. 

Such questions may seem abstract, but they are in fact very important in practice. Above, our proof of convergence of Jacobi s method is based upon the assumed existence of a complete eigenvector basis for — A). 

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