Tensor eigenvector

Symmetrized Shielding Tensor Eigenvectors (Principal Axes)... 

Moment Tensors Decomposition and Visualization, Fig. 6 Riedesel-Jordan source-type plot, (a) The original compact plot proposed by Riedesel and Jordan (1989) displaying the orientation of the moment tensor eigenvectors (P, T, and N axes), basic source types (ISO, CLVD, DC), and the position of the studied moment tensor (red dot), (b, c, d) A modified Riedesel-Jordan plot proposed... 

A better estimate of the shape of the polymer molecules, since they are highly anisotropic, is a representation of each molecule in terms of an equivalent spheroid with the same moment of inertia [45,46]. This is achieved by diagonalizing the moment of inertia tensor to obtain the eigenvectors a, b, and c and the principal moments 7, I/,/, and Icc. The moment of inertia tensor of molecule j is given by... 

NMR Chemical Shift Tensor Components eor Mes3Si /HCBiiMe5Br6 and Calculated Tensor Components eor MessSi", 1. " The Orientation oe the Eigenvectors Relative to the Molecular Frame is Given in Fig. 1... 

Kuz min et al. (15) pointed out a standard result of classical mechanics If a configuration of particles has a plane of symmetry, then this plane is perpendicular to a principal axis (19). A principal axis is defined to be an eigenvector of the inertial tensor. Furthermore, if the configuration of particles possesses any axis of symmetry, then this axis is also a principal axis, and the plane perpendicular to this axis is a principal plane corresponding to a degenerate principal moment of inertia (19). 

The principal strain rates are eigenvalues of the strain-rate tensor (matrix). As described more fully in Section A.21, the direction cosines that describe the orientation of the principal strain rates are the eigenvectors associated with the eigenvalues. In solving practical fluids problems, there is rarely a need to determine the principal strain rates or their orientations. Rather, these notions are used theoretically with the Stokes postulates to form general relationships between the strain-rate and stress tensors. It is perhaps worth noting that in solid mechanics, the principal stresses and strains have practical utility in understanding the behavior of materials and structures. 

Square matrices and tensors can be characterized by their eigenvalues and eigenvectors. If M is an n x n square matrix (or tensor), there is a set of n special vectors, e, each with its own special scalar multiplier A for which matrix multiplication of a vector is equivalent to scalar multiplication of a vector ... 

A rank-two property tensor is diagonal in the coordinate system defined by its eigenvectors. Rank-two tensors transform like 3x3 square matrices. The general rule for transformation of a square matrix into its diagonal form is... 

In this application of the BWR theory, Hudson and Lewis assume that the dominant line-broadening mechanism is provided by the modulation of a second rank tensor interaction (i.e., ZFS) higher rank tensor contributions are assumed to be negligible. R is a 7 X 7 matrix for the S = 7/2 system, with matrix elements written in terms of the spectral densities J (co, rv) (see reference [65] for details). The intensity of the i-th transition also can be calculated from the eigenvectors of R. In general, there are four transitions with non-zero intensity at any frequency, raising the prospect of a multi-exponential decay of the transverse magnetization. There is not a one-to-one correspondence between the... 

The eigenanalysis of the MIL tensor is run via Jacobi method to calculate the main characteristics values, that is, eigenvalues (eo-g), and characteristic directions, that is, eigenvectors (co-s)-... 

An efficient method to visualize the properties of the normal coordinates is to calculate activity measurements, AM. They show which components of the dipole moment vector and the polarizabilty tensor are modulated by the vibration, and the relative sign of the infrared and Raman optical activity (Schrader et al., 1984 Schrader, 1988). The necessary transformation of the eigenvectors (Eq. 5.2-13) needs only seconds of computer time. The AMs are useful to assign vibrations to symmetry species and to check the input of the frequency calculation the symmetry of the Cartesian coordinates of the atoms as well as of the force constant matrix. This program is part of the SPSIM program package (Fischer et al., 1989). 

Flow birefringence is due to the optical anisotropy created by the orientation of the macromolecules within the flow field. In a plane flow, if we denote by I and II the principal axes of the refractive index tensor (principal axes are those for which the tensor is diagonal they are defined by the eigenvectors of the tensor), the local birefringence A is defined as ... 

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