Eigenvector orthonormality

Note that eigenvector orthonormality can also be safely assumed for the degenerate case, cii = ap without loss of generality.)... 

We then allow Ri and R2 to vary, subject to orthonormality, just as in the closed-shell case. Just as in the closed-shell case, Roothaan (1960) showed how to write a Hamiltonian matrix whose eigenvectors give the columns U] and U2 above. 

Although the eigenvalues 0 and 1 are universal, there are many possible eigenvectors r that depend on the kind of states p is to designate. To each one of an orthonormal set of functions , for instance,... 

The same applies to the other eigenvectors U2 and Vj, etc., with additional constraints of orthonormality of u, U2, etc. and of Vj, Vj, etc. By analogy with eq. (31.5b) it follows that the r eigenvalues in A must satisfy the system of linear homogeneous equations ... 

The eigenvalue-eigenvector decomposition of a Hermitian matrix with the complete orthonormal set of eigenvectors Vi and eigenvalues A, is written as... 

Since H is Hermitian, the eigenvectors Vj of H form a complete orthonormal set and the vector representing a general state at t = 0 may be expressed as a linear superposition of these eigenvectors, (0) = CjVj, ... 

As indicated in Table 4.2, the eigenvalues of the Hessian matrix of fix) indicate the shape of a function. For a positive-definite symmetric matrix, the eigenvectors (refer to Appendix A) form an orthonormal set. For example, in two dimensions, if the eigenvectors are Vj and v2, v[v2 =0 (the eigenvectors are perpendicular to each other). The eigenvectors also correspond to the directions of the principal axes of the contours of fix). 

It is well established that the eigenvalues of an Hermitian matrix are all real, and their corresponding eigenvectors can be made orthonormal. A special case arises when the elements of the Hermitian matrix A are real, which can be achieved by using real basis functions. Under such circumstances, the Hermitian matrix is reduced to a real-symmetric matrix ... 

SVD is completely automatic. It is one of the most stable algorithms available and thus can be used blindly. It is one command in Matlab [U, S, Vt]=svd (Y, 0). The matrices U and V1 contain as columns so-called eigenvectors. They are orthonormal (see Orthogonal and Orthonormal Matrices, p.25) which means that the products... 

To be able to represent the spectral vectors in the plane, we need a system of axes, preferably an orthonormal system. As it turns out, the two eigenvectors V form an orthonormal system of axes in that plane. This is represented in Figure 5-11. 

Figure 5-11. The eigenvectors V form a system of orthonormal axis in the plane spanned by the spectra.

Due to the hermitian character of the dynamical matrix, the eigenvalues are real and the eigenvector satisfies the orthonormality and closure conditions. The coupling coefficients are given by... 

The elements of D represent the sum over all unit cells of the interaction between a pair of atoms. D has 3n x 3n elements for a specific q and j, though the numerical value of the elements will rapidly decrease as pairs of atoms at greater distances are considered. Its eigenvectors, labeled e ( fcq), where k is the branch index, represent the directions and relative size of the displacements of the atoms for each of the normal modes of the crystal. Eigenvector ejj Icq) is a column matrix with three rows for each of the n atoms in the unit cell. Because the dynamical matrix is Hermitian, the eigenvectors obey the orthonormality condition... 

What about the two degenerate eigenvectors v(l) and v(2) Are they also orthonormal So far, we know that these two eigenvectors have the structure... 

In all cases then, one can find n orthonormal eigenvectors (remember we required... 

Let cp, ..., cp m be the orthonormal system composed of the eigenvectors of R associated with the eigenvalue X. We assume that they all belong to the domain of R° . Then we can construct the Hermitian matrix... 

We denote by Aj., the system of eigenvalues and orthonormal eigenvectors of the operator A ... 

If A is a Hermitian matrix, then as proved above, the eigenvectors (n) can be chosen to be orthonormal. Hence [Equation (2.24)] the X matrix can be chosen as unitary therefore, X has an inverse (namely, its Hermitian conjugate X1). Application of X-1 on the left of each side of... 

Since F is a Hermitian operator, F is a Hermitian matrix and its eigenvectors can be chosen as orthonormal. Let C be the unitary square matrix of column eigenvectors of F ... 

Davidson introduced a different method for higher eigenvalues which also avoids the need to have the elements of H stored in any particular order. In this method the kill eigenvector of H for the ni iteration is expanded in a sequence of orthonormal vectors bi, i=l n with coefficients found as the k— eigenvector of the small matrix H with elements bTHBj. Convergence can be obtained for a reasonably small value of n if the expansion vectors b are chosen appropriately. Davidson defined... 

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