Tensor characteristic equation

A second-order real-valued symmetric tensor T gives three real eigenvalues A which are determined by the characteristic equation ... 

Like vectors, tensors of any order possess invariants, quantities that do not change during a coordinate transformation. A second-order tensor has three invariants, called its eigenvalues (from the German word eigen meaning own, peculiar, particular ) For a tensor (Ay), they can be calculated from the characteristic equation... 

The roots of Eq. (2A.4), often referred to as the characteristic equation of T, are called the principal values of T. Two cases are possible either (1) one root is real or (2) all three are real. If the tensor T is symmetric, then all principal values (A.j, X2, X3) are real. Suppose that all principal values (Xj, X2, and A.3) are real and distinct, which implies that the tensor T is symmetric. Then for each of the principal values, Eq. (2A.1) must be satisfied, that is. 

The Cayley-Hamilton theorem states that if the characteristic equation of a symmetric second-order tensor A is C(X) = 0, then the tensor A satisfies the equation C(A) = 0... 

The eigenvalues A of the image structure tensor can be used to detect lines, corners or constant grey value regions. The characteristic equation of matrix M is... 

The solutions of this characteristic equation are the eigenvalues of tensor... 

Structure Tensors in Seismic Data Analysis 65 The characteristic equation of the above matrix is the cubic polynomial... 

High-field EPR (HFEPR) spectroscopy greatly improves the resolution of the EPR signals for spectral features such as the g-tensor. Deviations of the g-value from free electron g=2.0023 are due to spin-orbital interactions, which are one of the most important structural characteristics (Kevan and Bowman 1990). Using a higher frequency results in enhanced spectral resolution in accordance with the resonance equation ... 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

However, in the earlier times, the constitutive relation for a viscoelastic liquid were formulated when the equations for relaxation processes could not be written down in an explicit form. In these cases the constitutive relation was formulated as relation between the stress tensor and the kinetic characteristics of the deformation of the medium (Astarita and Marrucci 1974). 

To calculate characteristics of linear viscoelasticity, one can consider linear approximation of constitutive relations derived in the previous section. The expression (9.19) for stress tensor has linear form in internal variables x"k and u"k, so that one has to separate linear terms in relaxation equations for the internal variables. This has to be considered separately for weakly and strongly entangled system. 

Thus, one can see that the single-mode approximation allows us to describe linear viscoelastic behaviour, while the characteristic quantities are the same quantities that were derived in Chapter 6. To consider non-linear effects, one must refer to equations (9.52) and (9.53) and retain the dependence of the relaxation equations on the anisotropy tensor. 

Equation (10.6), formulated in the previous section, defines the relative permittivity tensor in terms of the mean orientation of certain uniformly distributed anisotropic elements, which we shall interpret here as the Kuhn segments of the model of the macromolecule described in Section 1.1. We shall now discuss the characteristic features of a polymer systems, in which the segments of the macromolecule are not independently distributed but are concentrated in macromolecular coils. 

Equation (2.26) for heat conduction and Eq. (2.3) for momentum transfer are similar, and the flow is proportional to the negative of the gradient of a macroscopic variable the coefficient of proportionality is a physical property characteristic of the medium and dependent on the temperature and pressure. In a three-dimensional transport, Eqs. (2.27) and (2.15) differ because the heat flow is a vector with three components, and the momentum flow t is a second-order tensor with nine components. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

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