Eigenvectors diagonalization transformation

As a check, we verify that a similarity transformation with the unitary square matrix of eigenvectors diagonalizes Sx ... 

Relation Between Eigenvectors and Diagonalization Transformations Appendix B Coherent States in the Floquet Representation... 

A unitary transformation is then introduced whieh diagonalizes the FG matrix, yielding eigenvalues s, and eigenvectors q,. The kinetic energy operator is still diagonal in these eoordinates. 

The covariance matrix is factored using the diagonal matrix A and the eigenvector matrix U as U UT. Since 5 is symmetric and positive-definite, the eigenvalues are positive and the eigenvectors orthogonal. The inverse S 1 of S can be expanded as UA 1UT and the transformation... 

The 1-RDM can be diagonalized by a unitary transformation of the spin orbitals (jf), (x) with the eigenvectors being the natural spin orbitals (NSOs) and the eigenvalues , representing the ONs of the latter. 

When the parameters are strongly correlated, it is still possible to define a set of mutually uncorrelated combinations of the parameters. This can be shown as follows. If T represents the matrix of the eigenvectors of the variance-covariance matrix Mx, then Mx is diagonalized by the transformation... 

This equation can be solved several ways [40]. One method involves diagonalizing the Liouville matrix, iL + R+ K. The matrix, iL + R + K, is precisely the matrix that Binsch deals with [22, 35]. If U is the matrix with the eigenvectors as columns, and A is the diagonal matrix with the eigenvalues down the diagonal, then we can write (11) as (12). This is similar to other eigenvalue problems in quantum mechanics, such as the transformation to normal co-ordinates in vibrational spectroscopy. 

The matrices g and A transform the normal modes with frequencies Vj and eigenvectors V into atomic displacements (T), ADPs are the 3 x 3 diagonal blocks of and is a temperature-independent term accounting for the high-frequency vibrations (internal modes). 

This identity illustrates a so-called similarity transform of M using S. Since A is diagonal, we say that the similarity transform S diagonalizes M. Notice this would still work if we had used the eigenvectors v(k) in a different order. We would just get A with the diagonal elements in a different order. Note also that the eigenvalues of M are the same as those of... 

The matrix Q can now be transformed into a stochastic matrix, which will be descriptive of the restricted random walks rather than of their generation employing probabilities based on unrestricted walk models. The transformation is performed as follows Let Xt be the largest eigenvalue of the matrix Q, and let Sj be the corresponding left-hand side eigenvector (defined by SjQ = X ). Let A be a diagonal matrix with elements a(i,j) = (/) 8st = [ 1(1),. v,(2),..., (v)] and 8(i,j) is the... 

If X is the matrix formed from the eigenvectors of a matrix A, then the similarity transformation X lAX will produce a diagonal matrix whose elements are the eigenvalues of A. Furthermore, if A is Hermitian, then X will be unitary and therefore we can see that a Hermitian matrix can always be diagonalized by a unitary transformation, and a symmetric matrix by an orthogonal transformation. 

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