Tensor transformation matrix

Another method to determine the magnitude and rhombicity of the alignment tensor is based on the determination of the Saupe order matrix. The anisotropic parameter of motional averaging is represented by this order matrix, which is diagonalized by a transformation matrix that relates the principal frame, in which the order matrix is diagonal,... 

X, ..., X. Such a transformation induces a trivial tensor transformation for the instantaneous force Tip(t). We show in the Appendix, Section H, by evaluating the time and ensemble average of the instantaneous force over a short time interval, that, in the case of a nonsingular mobility matrix, such a transformation creates a transformed force bias... 

It should be noted that the metric factors represent diagonal elements of a transformation matrix. It is therefore prudent to check the off-diagonal components to ensure that the new coordinate system is indeed orthogonal. In general, the elements of the metric tensor are given as [257]... 

Referring to Fig. A.2, assume that the principal coordinates align with z, r, and O. The unit vectors (direction cosines) just determined correspond with the row of the transformation matrix N. Thus, if the principal stress tensor is... 

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... 

The bond polarization model gives the chemical shift of an atom a as the sum over the Na bonds of this bond. The bond contributions are formed of a component for the unpolarized bond (which also includes the inner shell contributions to the magnetic shielding) and a polarization term. The bond contributions are represented by a tensor with its principal axes along the basis vectors of the bond coordinate system. The transformation from the bond coordinate system into a common cartesian system is given by the transformation matrix Z). ... 

Guha and Chase reported that a Raman spectrum could be observed only in experiments that select a component of the polarizability tensor transforming as Ee or Ee [1]. The electronic matrix elements in these polarization geometries are [1]... 

F Transformation matrix for linear and circular basis sets, (1.57) strain tensor, (7.41). 

Equation 10.1 is a second-rank tensor with transpose symmetry. The normal components of stress are the diagonal elements and the shear components of stress are the nondiagonal elements. Although Eq. 10.1 has the appearance of a [3 x 3] matrix, it is a physical quantity that, for one set of axes, is specified by nine components, whereas a transformation matrix is an array of coefficients relating two sets of axes. The tensor coefficients determine how the three components of the force vector, /, transmitted across a small surface element, vary as different values are given to the components of a unit vector / perpendicular to the face (representing the face orientation) ... 

The basis of the principal frame is the eigenvectors of the inertia tensor. The matrix that transforms a vector from the space fixed frame to the principal frame is denoted by A,-, the rows of this matrix are the vectors (u,-,v,-,w,). The inverse of A,- transforms a vector from the principal frame to the space fixed frame is simply Af. 

The decoupled representation involves the normal population modes defined by the columns of the transformation matrix, U = f4, which diagonalizes the AIM hardness matrix, i.e., rotates the coordination system to the principal axes of the hardness tensor [33-36, 44] ... 

Tensors of 2nd order A are linear transformations (matrix 3 x 3) of vector a to vector b... 

The <, Uf, Uy can be thought of as polar vector components (as opposed to axial vector components u, Uy, ) and they transform accordingly. When the lattice dynamical problem is treated in terms of the dynamical variable ujtyU ujigUy, Cochran and Pawley have pointed out that the two-molecule interaction force constants 0, (/A , I k ) can be treated as a two-dimensional tensor of dimension six. If S is the cartesian coordinate transformation matrix corresponding to a symmetry transformation, then the six-dimensional transformation matrix is... 

Notice that the terms r j of the submatrix R are functions of the angles between the axes of two reference frames, so fhe acfual independenf variables are three and not nine. The notation of Equafion 1.1 means fhat T is the tensor transforming poses from reference frame PI to reference frame P2. Tensors can be inverted to reverse the transformation and chain multiplied to combine different transformations. In the multiplications the right upper index of the first matrix must equal the lower right index of the second. 

The effect of the rotations as described by A R) and A RJ) on the Lorentz boost matrix A v) is as follows the element A°q remains unaffected, the elements M g and AP I transform like a 3-vector, and the elements A transform like a 3-tensor of second rank. Since the transformation matrix A( ) must reduce to the familiar Lorentz boost in c-direction as given by Eq. (3.66) for V = vei, we must have... 

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