Generalized column-eigenvectors

In the column-problem we compute the matrix of generalized column-eigenvectors B and the diagonal matrix of eigenvalues from ... 

The SVD is generally accepted to be the most numerically accurate and stable technique for calculating the principal components of a data matrix. MATLAB has an implementation of the SVD that gives the singular values and the row and column eigenvectors sorted in order from largest to smallest. Its use is shown in Example 4.3. We will use the SVD from now on whenever we need to compute a principal component model of a data set. 

The general formofthe eigenvector decomposition (3.89) is JT= WA, where IFis amatrix whose column vectors are eigenvectors of A. Therefore, for any normal matrix A, we can form a unitary matrix whose coliunn vectors are eigenvectors to write A in Jordan normal form. 

We will start out by assuming that S = 1 and leave the more general case until later. When the entire eigensystem is required, the eigenvectors are collected as columns into a matrix U, and the eigenvalues A as diagonal elements into a diagonal matrix D. The equation then takes the form... 

The expansion column matrix A from Eq. (26) is the eigenvector of U(s), and the elements Snm of the overlap matrix S are given in Eq. (19). The obtained expression (26) is not an ordinary but a generalized eigenvalue problem involving the overlap matrix S due to the mentioned lack of orthogonality... 

The general method of solution that was proposed by Toor and by Stewart and Prober exploits the properties of the modal matrix [P] whose columns are the eigenvectors of [7)] (see Appendix A.4). The matrix product... 

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