Eigenvector generalized

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

This equation determines a rank-1 matrix, and the eigenvector of its only one nonzero eigenvalue gives the direction dictated by the nonadiabatic couphng vector. In the general case, the Hamiltonian differs from Eq.(l), and the Hessian matrix has the form... 

We have presented above the derivation of eqns 38 and 39 in great detail because it includes expressions of general utility, in particular the variation of the eigenvectors (eqns 7 and 24) of an MO problem after Lowdin orthogonalization and the resulting variation of the population matrix P. The generalization to a Hamiltonian more complicated than that of eqn 19 is possible by following step by step the above derivation. 

Thus far we have considered the eigenvalue decomposition of a symmetric matrix which is of full rank, i.e. which is positive definite. In the more general case of a symmetric positive semi-definite pxp matrix A we will obtain r positive eigenvalues where r general case we obtain a pxr matrix of eigenvectors V such that ... 

The transposed product XX has the same r eigenvalues in Is , but generally possesses r different eigenvectors in U ... 

Eigenvector projections are those in which the projection vectors u and v are eigenvectors (or singular vectors) of the data matrix. They play an important role in multivariate data analysis, especially in the search for meaningful structures in patterns in low-dimensional space, as will be explained further in Chapters 31 and 32 on the analysis of measurement tables and general contingency tables. 

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

An instructive description of the first-order perturbation treatment of the quadrupole interaction in Ni has been given by Travis and Spijkerman [3]. These authors also show in graphical form the quadrupole-spectrum line positions and the quadrupole-spectrum as a function of the asymmetry parameter r/ they give eigenvector coefficients and show the orientation dependence of the quadrupole-spectrum line intensities for a single crystal of a Ni compound. The reader is also referred to the article by Dunlap [15] about electric quadrupole interaction, in general. 

If A has repeated eigenvalues (multiple roots of the characteristic polynomial), the result, again from introductory linear algebra, is the Jordan canonical form. Briefly, the transformation matrix P now needs a set of generalized eigenvectors, and the transformed matrix J = P 1 AP is made of Jordan blocks for each of the repeated eigenvalues. For example, if matrix A has three repealed eigenvalues A,j, the transformed matrix should appear as... 

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