Eigenvectors, theorems

At this stage, we are ready to prove that Kramers theorem holds also for the total angular momentum F. We will do it by reductio ad absurdum. Then, let Itte) be the eigenvector of H with eigenvalue E,... 

The secular problem, in either form, has as many eigenvalues Ei and eigenvectors Cij as the dimension of the Hu matrix as . It can also be shown that between successive pairs of the eigenvalues obtained by solving the secular problem at least one exact eigenvalue must occur (i.e., Ei+i > Egxact > Ei, for all i). This observation is referred to as the bracketing theorem. ... 

A theorem, which we do not prove here, states that the nonzero eigenvalues of the product AB are identical to those of BA, where A is an nxp and where B is a pxn matrix [3]. This applies in particular to the eigenvalues of matrices of cross-products XX and X which are of special interest in data analysis as they are related to dispersion matrices such as variance-covariance and correlation matrices. If X is an nxp matrix of rank r, then the product X X has r positive eigenvalues in A and possesses r eigenvectors in V since we have shown above that ... 

Stokes theorem, geometric phase theory, eigenvector evolution, 14-17 Stueckelberg oscillations, direct molecular dynamics, trajectory surface hopping, 398-399... 

However, it is easy to verify that neither %a nor %b is an eigenvector of this unperturbed Hamiltonian, and neither are ea and eb its eigenvalues (see Example 3.18). More generally, since H/(0) is clearly Hermitian, it cannot have any non-orthogonal eigenvectors, by virtue of the theorem (note 79) quoted above. 

The famous Gershgorin theorem gives estimates of eigenvalues. The estimates of correspondent eigenvectors are not so well-known. In the chapter we use some estimates of eigenvectors of kinetic matrices. Here we formulate and prove these estimates for general matrices. Below A = (a,y) is a complex n x n matrix. Pi — (sums of nondiagonal elements in rows), Qi — (sums of... 

The reader may wish to compare this Spectral Theorem to Proposition 4.4. Proof. To find the eigenvalues of A, we consider its characteristic polynomial. Then we use eigenvectors to construct the matrix M. 

To show this we first consider [Pg.40]

Theorem. If X is the matrix formed from the eigenvectors of A, then X XAX is a diagonal matrix A composed of the eigenvalues of A. 

The other theorem states that the matrix X formed by using the eigenvectors of a Hermitian matrix as its columns is unitary (for the definition of a unitary matrix, see Appendix A.4-l(ff)). The proof of these two theorems is given in Appendix A.4-3. 

Theorem. If A is Hermitian, its eigenvalues are real and its eigenvectors are orthogonal to each other provided they correspond to non-degenerate eigenvalues. 

The deep connection between A and its spectrum of eigenvalues at and eigenvectors y i is best exhibited by the spectral theorem ... 

The spectral theorem can also be used to express many functions of A, by recognizing that all powers of A have the same eigenvectors as A and the associated eigenvalues are equivalent functions of the a . 

Exercise. Prove the following theorem 510. The macroscopic equation is linear if and only if the function Q[y) = y— [Pg.127]

Some real matrix classes, studied in subsection (F) below, however, have only real eigenvalues and corresponding real eigenvectors. The complication with complex eigenvalues and eigenvectors is caused by the Fundamental Theorem of Algebra which states that all the roots of both real and complex polynomials can only be found in the complex plane C. 

For a space of eigenvectors of matrices of the gaussian orthogonal ensemble (k = N) the distribution of values of matrix elements of electromagnetic transition operators is gaussian, as follows from the central limit theorem. The ensemble averaging of hamiltonians guarantees that no correlations exist between the hamiltonian structure and the particular transition operator that is considered. 

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