The logarithmic matrix function

In this section, we introduce the logarithm, log A, of a nonsingular matrix A, required to satisfy the relationship 

To simplify our discussion of log A, we restrict the domain of the logarithmic matrix function to nonsingular matrices A that can be diagonalized 

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The logarithmic matrix function log A is thus the true inverse function of the exponential expA in the neighbourhood of 1, mapping matrices in the neighbourhood of 1 onto matrices in the neighbourhood of 0. 

The following properties of the logarithmic matrix function are proved in Exercise 3.6 ... 

Prove the following relations for the logarithmic matrix function ... 

Since A, B and AB belong to the domain where the exponential matrix function is the inverse of the logarithmic matrix function, we obtain... 

In 1973 Andersen and Woolley [1.25] extended the LCMTO method to molecular calculations. At the end of their paper they introduced that choice of MTO tail, i.e. proportional to J = 9i/j/9E, which in a natural fashion ensured orthogonality to the core states and at the same time led to an accurate and elegant formulation of linear methods. The resulting, technique was immediately developed in a paper by Andersen [1.26] which, in a condensed form, contains most of what one need know about the simple concepts of linear band theory. Thus, we find here the KKR equation within the atomic-sphere approximation at this stage is called ASM the LCMTO secular matrix, latter called the LMTO matrix the energy-independent structure constants and the canonical bands and the Laurent expansion of the logarithmic-derivative function and the corresponding potential parameters. 

Fig. 3.1. The relationship between the exponential and logarithmic matrix functions. The domains for one-to-one mappings in the neighbouihoods of 0 and 1 are indicated by circles.

Fig. 31.14. Performance of three computer algorithms for eigenvalue decomposition as a function of the dimension of the input matrix. The horizontal and vertical scales are scaled logarithmically. Execution time is proportional to a power 2.6 of the dimension.

It is evident that this potential leads to a logarithm squared contribution of order a (Za) after substitution in (3.71). One may obtain one more logarithm from the continuous spectrum contribution in (3.71). Due to locality of the potential, matrix elements reduce to the products of the values of the respective wave functions at the origin and the potentials in (3.72). The value of the continuous spectrum Coulomb wave function at the origin is well known (see, e.g., [94]), and... 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

The theory of the / -matrix can be understood most clearly in a variational formulation. The essential derivation for a single channel was given by Kohn [202], as a variational principle for the radial logarithmic derivative. If h is the radial Hamiltonian operator, the Schrodinger variational functional is... 

This is the reciprocal of the logarithmic derivative of the wave function Xdiq) in the dissociation channel, for q = qd. At given total energy E these R-matrix elements are matched to external scattering wave functions by linear equations that determine the full scattering matrix for all direct and inverse processes involving nuclear motion and vibrational excitation. Because the vibronic R-matrix is Hermitian by construction (real and symmetric by appropriate choice of basis functions), the vibronic 5-matrix is unitary. 

Berens [7] reported that under such conditions the logarithm of diffusivity is a linear function of the molecular diameter of the penetrant molecule. This is consistent with the hypothesis discussed in detail by Stern and Frisch [8, 9] that diffusion through a rigid polymeric matrix is proportional to the energy required to expand (i.e. swell temporarily on a molecular scale) the polymeric chains sufficiently to allow peristaltic motion of the penetrant molecules along these chains. This energy is presumed to be proportional to the molecular diameter of the penetrant. 

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