Rotation-reflection transformation matrices

The major problem is to find the rotation/reflection which gives the best match between the two centered configurations. Mathematically, rotations and reflections are both described by orthogonal transformations (see Section 29.8). These are linear transformations with an orthonormal matrix (see Section 29.4), i.e. a square matrix R satisfying = RR = I, or R = R" . When its determinant is positive R represents a pure rotation, when the determinant is negative R also involves a reflection. 

Rotation-reflection. Consider a rotation by 0 (— 2w/n) about the e, base veotor, followed by reflection in the cTu plane. The components of the point vector p (or, the coordinates of the point P) will be first transformed by the rotation, as in (1), and then these new components (coordinates) will be transformed by the reflection, as in (2). Using matrix notation, these two transformations can be combined into one step (see 4-3(3)) and we get... 

For a particular material response or applied field, particular choices of coordinate axis orientations may be especially convenient (e.g., axes aligned with crystal lattice vectors). Linear transformations—such as rotations, reflections, and affine distortions— can be performed on vector forces and responses by matrix multiplication to describe force-response relations in different coordinate systems. For instance, a vector E can be transformed between old and new coordinate systems by a matrix A ... 

Note that classical PCA is not affine equivariant because it is sensitive to a rescaling of the variables. But it is still orthogonally equivariant, which means that the center and the principal components transform appropriately under rotations, reflections, and translations of the data. More formally, it allows transformations XA for any orthogonal matrix A (that satisfies A-1 = A7). Any robust PCA method only has to be orthogonally equivariant. 

In addition, the determinant of the a s is unity. The matrix formed by a set of a s which satisfy the relations 10 7 is said to be a unitary matrix the matrices representing rotations, reflections, and inversions are unitary. It will be noted that the matrices 10 3 and 104 satisfy the relations 10 7. The transformation which is the reverse of 10 6 is given by the set of equations... 

It must be emphasized that Procrustes analysis is not a regression technique. It only involves the allowed operations of translation, rotation and reflection which preserve distances between objects. Regression allows any linear transformation there is no normality or orthogonality restriction to the columns of the matrix B transforming X. Because such restrictions are released in a regression setting Y = XB will fit Y more closely than the Procrustes match Y = XR (see Section 35.3). 

For the special case of S2 s i, the mirror image is produced by the inversion operation, but must be rotated by 180° to bring it into an exact reflective relationship to the original. This can be seen in Figure 3.4 and is conveniently expressed by using the matrices for the coordinate transformations. (Readers unfamiliar with matrix algebra may consult Appendix I.) Thus, we represent the operation S2 35 i hy the first matrix shown below and a rotation by n... 

This pattern—a rank-one tensor is transformed by a single matrix multiplication and a rank-two tensor is transformed by two matrix multiplications—holds for tensors of any rank. If A is an orthogonal transformation, such as a rigid rotation or a rigid rotation combined with a reflection, its inverse is its transpose. For example, if R is a rotation, RijRji = 8, where 5 is the Kronecker delta, defined as... 

Let us set up a 2D unitary matrix representation for the transformation of the spin functions a and (1 in Civ. So far, we have established only a relation between 0(3)+ and SU(2). The matrix representations of reflections or improper rotations do not belong to 0(3)+ because their determinants have a value of -1. To find out how a and p behave under reflections, we notice that any reflection in a plane can be thought of as a rotation through n about an axis perpendicular to that plane followed by the inversion operation. For instance, 6XZ may be constructed as xz = Cz(y) i. Herein, it is not necessarily required... 

The effect of applying two sequential coordinate transformations on a point, r, can be represented by the product of the two matrices, each one of which represents the respective transformation. We need to take care, however, that the matrices are multiplied in the correct order because, as we saw above, matrix multiplication is often non-commutative. For example, in order to find the matrix representing an anticlockwise rotation by 0, followed by a reflection in the y-axis, we need to find the product CA (and not AC as we might initially assume ). 

If the matrix S is an orthogonal matrix (S S = SS = I), then the transformation is called an orthogonal transformation. An orthogonal transformation is a reflection and/or a rotation. The property, as exemplified in and below Equation (5.15), is called rotational freedom and although this term is rather sloppy, it is commonly used. 

A rotoreflection improper rotation) S about an axis is a rotation by the angle 0 followed by reflection by the plane perpendicular to the axis. This is neither a pure rotation nor a pure reflection, but a combined operation. It transforms a left hand into a right hand and vice-versa. This is called an operation of the second type. The determinant of any matrix representing an operation of the... 

The matrices Q and U are obtained in the procedure of reduction of the matrix A using a series of orthogonal transformations, like Householder reflections (Householder 1958 Wilkinson 1965 George and Ng 1986) or Givens rotations (Givens 1954 Wilkinson 1965, George and Heath 1980), on the upper triangular form. 

Reflections of a rotational vector in any mirror plane that contains that vector will therefore cause the vector to rotate in the opposite (or clockwise) direction, transforming it into the opposite of itself, or a matrix of [—1]. The representations for the three rotational vectors shown in Figure 8.12 are listed in the table. 

The canonical MOs obtained as eigenfunctions of the Fock operator have contributions from all basis functions and thus are delocalized over all centers. They do not reflect the common picture of localized bonds between two atoms. However, the orbitals may be freely transformed by making linear combinations without changing the total wave function. Hence, an orbital rotation matrix can be applied to transform the canonical into localized orbitals which reflect bonds between two atoms. Several localization schemes were proposed, but the natural bond... 

Iterated Function Systems. Iterated function systems make it possible to re-create natural objects using mathematical descriptions collected into data sets with accompanying rules for computation. This process is also known as fractal image compression, and it is the technology that makes computer-generated landscapes and visual effects in film possible. Very simply, a table of numbers (or matrix formulation) is created describing the affine transformations desired. These follow the conventional order of scalings, reflections, rotations, and translations. 

An improper rotation is a rotation followed by a reflection in a plane perpendicular to the axis of rotation. An example is the S2 symmetry operation. The inversion through a center of symmetry is the same as an improper rotation through an angle of 180°. A reflection by a plane of symmetry is an improper rotation through an angle of zero degrees. The transformation equations for an improper rotation are the same as those for a proper rotation except that z = — z. Equation (3.33) can be written in matrix form as... 

As a rotation transforms one orthogonal basis into another, a matrix satisfying (3.8) is said to be orthogonal. Note that matrices that perform improper rotations, i.e. that combine a rotation with a mirror reflection, also map orthogonal bases into other ones, and also satisfy... 

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