Orthogonal eigenvectors

A self-adjoint operator A in the space Rn possesses n mutually orthogonal eigenvectors, ... We assume that all the j. s are normalized, that is, Mi II = 1 for k = I,..., n. Then ( j, i,) = The corresponding eigenvalues are ordered with respect to absolute values ... 

We may construct an n X n matrix X using the n orthogonal eigenvectors x as columns... 

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

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. 

A self-adjoint operator A in the space Rn possesses n mutually orthogonal eigenvectors, n. We assume that all the j. s are nor ... 

This empirical statistical function, based on the residual standard deviation (RSD), reaches a minimum when the correct number of factors are chosen. It allows one to reduce the number of columns of R from L to K eigenvectors or pure components. These K independent and orthogonal eigenvectors are sufficient to reproduce the original data matrix. As they are the result of a mathematical treatment of matrices, they have no physical meaning. A transformation (i.e. a rotation of the eigenvectors space) is required to find other equivalent eigenvectors which correspond to pure components. 

It is seen that any (n x k) matrix can be factorized into a (n x k) score matrix T and an orthogonal eigenvector matrix. The elements of column vectors t in T are the score values of the compounds along the component vector p . Actually, the score vectors are eigenvectors to XX. ... 

Des Cloizeaux has shown that one can explicitly construct a similarity transformation of 9if which leaves the eigenvalue spectrum invariant, but enforces hermiticity and therefore orthogonal eigenvectors for distinct eigenvalues,... 

This shows that any real symmetric matrix can be factorized into orthogonal eigenvector matrices and a diagonal matrix of the corresponding eigenvalues." ... 

The row vectors in matrix Vred define points in the three-dimensional space of the orthogonal eigenvectors. They lie in (or near, in the presence of noise) a plane, consistent... 

S is a symmetric matrix [since correlation Xi.Xj) - correlation Xj,Xi)] of real values. Therefore, from the theory of eigenanalysis, it follows that P is the matrix whose columns are the orthogonal eigenvectors of S and that, ..., are the corresponding eigenvalues. For a more detailed discussion of eigensystem theory see, for example, Morris (1982). 

Thus, in equilibrium, the Descartes coordinates of the atom V are (xy,yy, Zy) where Xy, yy andzv are the v-th components in order of the vectors X, Y and Z. The matrix elements of the matrix W are calculated at the equilibrium position of the atoms. If the centre of mass of the molecule is in the origin and the molecule is directed in such a way that the eigenvectors of its tensor of inertia are showing to the directions of the x, y and z axis, then the vectors X, Y and Z are orthogonal eigenvectors of the matrix W. From the construction of the matrix elements follow that WU = 0 if Mv = If the centre of mass is the origin of... 

The strength of the individual singlecouple and vector-dipole Green s function solutions (G j) is given by My which is a second-order tensor that is both real and symmetric (Mi2 = M21) enabling the decomposition into three orthogonal eigenvectors ... 

The matrix G(a) is symmetric since a is real. The eigenvalue problem (12.4.19) therefore has real eigenvalues and orthogonal eigenvectors. 

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