Matrix inverse square-root

VI. Finding Inverses, Square Roots, and Other Funetions of a Matrix Using its Eigenveetors and Eigenvalues... 

It is first transfonned to mass-dependent coordinates by a G matrix eontaining the inverse square root of atomic masses (note that atomic, not nuclear, masses are used, this is in line with the Bom-Oppenheimer approximation that the electrons follow the nucleus). 

Matrix containing square root of inverse atomic masses... 

The inverse matrix, B, is normalized by the reduced [Equation (13)] to give the variance-covariance matrix. The square roots of the diagonal elements of this normalized matrix are the estimated errors in the values of the shifts and, thus, those for the parameters themselves. These error estimates are based solely on the statistical errors in the original powder diffraction pattern intensities and can not accommodate the possible discrepancies arising from systematic flaws in the model. Consequently, the models used to describe the powder diffraction profile must accurately represent a close correspondence to... 

There is a unified procedure by which the canonical orthogonalization procedure can be extended to return normalized functions. Normalization results, if the canonical transformation involves the matrix of inverse square roots of the eigenvalues of the overlap matrix. This is the S procedure applied in modem molecular orbital programmes. 

The generation of the inverse square root of a matrix is a special case of the calculation of a general function of a matrix... 

A symmetrical orthogonalization corresponds to a transformation that has the property X SX = I, where X denotes all the x, coordinate vectors and S contains the overlap elements. One such transformation is given by the inverse square root of the overlap matrix (X = ), a procedure used in solving the self-consistent field equations in... 

Alternatively, a canonical orthogonalization can be performed by using the unitary matrix obtained by diagonalizing the overlap matrix and weighting by the inverse square root of the eigenvalues. 

If you will recall from the discussion of functions of a matrix in Chapter 1, we can form by diagonalizing S to form s, then taking the inverse square root of each of the eigenvalues to form the diagonal matrix s and then undiagonalizing by the transformation in (3.167). If S is Hermitian then is also Hermitian. Substituting (3.167) into (3.163),... 

Filled rubbers form a complex network of cross-linked chains connected to surface-active particles such as carbon black or amorphous silica (see Carbon Black). Here we will only indicate the structural features of importance in unfilled cross-linked elastomers. Two breakdown mechanisms are conceivable the initiation and growth of a cavity in a moderately strained matrix and the accelerating, cooperative rupture of interconnected, highly loaded network chains. The second mechanism is more important imder conditions, which permit the largest breaking elongation Xbmax to be attained (29). In that case, the quantity >-bmax is expected to be proportional to the inverse square root of the cross-link density Vg in fact, an increase of A,bmax with to is found experimentally for a... 

In the following paragraphs, the inverse and inverse square root of the overlap matrix of Eq. (7) is constructed. Clearly, it suffices to focus on of Eq. (6). We... 

This matrix is a diagonal matrix, its elements being the inverse square roots of the corresponding elements of D. With this definition, it is easy to see that... 

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