Properties of Eigenvalues and Eigenvectors

It is of interest to stress some properties hidden in the eigenvalues 

From the eigenvectors of Equations (1.23) we can construct the two square symmetric matrices of order 2  

The two matrices Pi and P2 do not admit inverse (the determinants of both are zero) and have the properties  

In mathematics, matrices having these properties (idempotency, mutual exclusivity, completeness3) are called projectors. In fact, acting on matrix C of Equation (1.21) 

This makes evident the projector properties of matrices Pi and P2. Furthermore, matrices Pi and P2 allow one to write matrix A in the so-called canonical form  

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Let us dwell on the properties of eigenvalues and eigenvectors of a linear self-adjoint operator A. A number A such that there exists a vector 0 with = A is called an eigenvalue of the operator A. This vector... 

PROPERTIES OF EIGENVALUES AND EIGENVECTORS Equation (1.44) is easily verified ... 

The evolution of this determinant first yields the eigenvalues. The solution of the whole eigenvalue problem provides pairs of eigenvalues and eigenvectors. The mathematical algorithm is described in detail in [MALINOWSKI, 1991]. A simple example, discussed in Section 5.4.2, will demonstrate the calculation. The following properties of these abstract mathematical measures are essential ... 

The following properties of the resulting eigenvalues and eigenvectors are of particular interest. 

In the following, we will consider for simplicity the case A] = — A2 so that 8 = — 2Ai. For frozen values of the two fields fij and 02, we calculate dressed states and dressed energies by diagonalizing Keff. The eigenelements can be labeled with two indices One, denoted n, refers to the levels of the atom and another one, denoted ft, refers to the relative photon numbers. The index k stands for the number of the coi photons absorbed and the number of 0)2 photons emitted. The eigenvalues and eigenvectors have the following property of periodicity ... 

A normal mode calculation is based upon the assumption that the energy surface is quadratic in the vicinity of the energy minimum (the harmonic approximation). Deviations from the harmonic model can require corrections to calculated thermodynamic properties. One way to estimate anharmonic corrections is to calculate a force constant matrix using the atomic motions obtained from a molecular d)namics simulation such simulations are not restricted to movements on a harmonic energy surface. The eigenvalues and eigenvectors are then calculated for this quasi-harmonic force-constant matrix in the normal way, giving a model which implicitly incorporates the anharmonic effects. 

In RUN, copies of these matrices are multipKed by a set of associated parameters and then summed. The total matrix is diagonalized and the ligand-field properties calculated from the resulting eigenvalues and eigenvectors. 

The bond critical point properties of an electron density distribution are evaluated at the bond critical point, rc, of a bonded interaction. Collectively, they consist of the curvatures and the Laplacian of the distribution, the value of /9(rc) and the bonded radii of the bonded atoms. The curvatures of p(rc) determine the local concentration or local depletion of the electron density distribution in the vicinity of the bond critical point measured in three mutually perpendicular directions. As observed by Bader and Essen (1984), the curvatures in these directions are found by evaluating the eigenvalues and eigenvectors of the Hessian matrix of p(rc), Hy = p(r l dXjdxj, (ij = 1,3). The three... 

The simplest spectroscopic property that can be computed by DFT is the vibrational density of states, which measures the response of the system to a periodic external perturbation coupled to the atomic (nuclear) coordinates. At T = 0 K this property is fully described by eigenvalues and eigenvectors of the dynamical matrix... 

Detailed properties of the eigenvalues and eigenvectors of (2.58) are given in Montroll and Goel, I. 

In the HF LCAO method, (4.57) for the periodic systems replaces (4.33) written for the molecular systems. In principle, the above equation should be solved at each SCF procedure step for all the (infinite) fe-points of the Brillouin zone. Usually, a finite set kj j = 1, 2,..., L) of fe-points is taken (this means the replacing the infinite crystal by the cyclic cluster of L primitive cells). The convergence of the results relative to the increase of the fe-points set is examined in real calculations, for the convergent results the interpolation techniques are used for eigenvalues and eigenvectors as these are both continuous functions of k [84]. The convergence of the SCF calculation results is connected with the density matrix properties considered in Sect. 4.3... 

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