Orthogonal rotation matrix

The columns of V are the abstract factors of X which should be rotated into real factors. The matrix V is rotated by means of an orthogonal rotation matrix R, so that the resulting matrix F = V R fulfils a given criterion. The criterion in Varimax rotation is that the rows of F obtain maximal simplicity, which is usually denoted as the requirement that F has a maximum row simplicity. The idea behind this criterion is that real factors should be easily interpretable which is the case when the loadings of the factor are grouped over only a few variables. For instance the vector f, =[000 0.5 0.8 0.33] may be easier to interpret than the vector = [0.1 0.3 0.1 0.4 0.4 0.75]. It is more likely that the simple vector is a pure factor than the less simple one. Returning to the air pollution example, the simple vector fi may represent the concentration profile of one of the pollution sources which mainly contains the three last constituents. 

Thus the problem of deducing E and C is the determination of the proper non-orthogonal rotation matrix, R. 

The orthogonal rotation matrix R(co,) which will perform the rotation using an angle set co is defined as ... 

R(6) is an orthogonal rotation matrix. The symbol tilde ( ) means here transposed and M(0) is the Mueller matrix of the given optical element corresponding to 0°. If e.g. a linear polarizer is rotated at an angle 6 the above transformation should look like as follows ... 

A rotation can be described by an orthogonal rotation matrix R. Orthogonality implies that R = r , i.e. R R = RR = I (unit matrix). Insertion of RR in the factor model gives... 

This transformation matrix has the same form as an orthogonal rotation matrix... 

Application of the collective relaxation model requires first comparing heteronuclear order parameters for C-H and N-H obtained experimentally (not necessarily a complete set) with those calculated from normal mode or quasi-harmonic normal mode analysis. The agreement between the calculated and experimentally determined order parameters can be improved by (I) a global scaling of the frequencies of the low frequency modes (2) adjustment of individual eigenfrequen-cies and (3) application of an orthogonal rotation matrix to mix the directions of the low frequency modes. Penalty functions are applied to restrict the magnitude of adjustment for the frequencies. 

These various techniques were recently applied to molecular simulations [11, 20]. Both of these articles used the rotation matrix formulation, together with either the explicit reduction-based integrator or the SHAKE method to preserve orthogonality directly. In numerical experiments with realistic model problems, both of these symplectic schemes were shown to exhibit vastly superior long term stability and accuracy (measured in terms of energy error) compared to quaternionic schemes. 

A linear coordinate transformation may be illustrated by a simple two-dimensional example. The new coordinate system is defined in term of the old by means of a rotation matrix, U. In the general case the U matrix is unitary (complex elements), although for most applications it may be chosen to be orthogonal (real elements). This means that the matrix inverse is given by transposing the complex conjugate, or in the... 

For instance, the first row of the matrix X defines a point with the coordinates (x, y,) in the space defined by the two orthogonal axes = (I 0) and = (01). Factor rotation means that one rotates the original axes = (1 0) and = (0 1) over a certain angle 9. With orthogonal rotation in two-dimensional space both axes are rotated over the same angle. The distance between the points remains unchanged. 

A section has been added to Chapter 1 on the distinction between analytic vs. enumerative studies. A section on mixture designs has been added to Chapter 9. A new chapter on the application of linear models and matrix least squares to observational data has been added (Chapter 10). Chapter 13 attempts to give a geometric feel to concepts such as uncertainty, information, orthogonality, rotatability, extrapolation, and rigidity of the design. Finally, Chapter 14 expands on some aspects of factorial-based designs. 

An alternative approach for selecting the number of retained factors may be found by examing the partition of variance after the orthogonal rotation. It can be argued that a factor with a variance of less than one contains less Information than did one of the original variables. However, since the objective of the rotation is to redistribute the variance from the artificially compressed state that results from the matrix diagonalization, it appears to be useful to examine a number of solutions with differing numbers of retained factors. The rotated solutions that contain factors with total variance less than one can then be rejected. For this example, the fine fraction results yield... 

The trace C matrix is, therefore, equal to a non-orthogonal rotation of QT... 

These criteria lead to different numeric transformation algorithms. The main distinction between them is orthogonal and oblique rotation. Orthogonal rotations save the structure of independent factors. Typical examples are the varimax, quartimax, and equi-max methods. Oblique rotations can lead to more useful information than orthogonal rotations but the interpretation of the results is not so straightforward. The rules about the factor loadings matrix explained above are not observed. Examples are oblimax and oblimin methods. 

Where X is an antisymmetric matrix containing the independent (orthogonal) rotation parameters. Expanding the energy in X about the origin... 

Four by Four. A convenient way of describing a symmetry operation is by using, not the 3x3 matrix that could represent three orthogonal rotations or three translations (but not both), but rather a 4 x 4 augmented matrix Q. For instance, we can represent the symmetry operator number 3, namely (—x + y, —x, z + 2/3) in space group P 3i 2 1 ( 152) as the matrix Q3 ... 

The first sum extends over all space group symmetry operators 0, ts) composed of a rotation matrix and a translation vector tg. The second sum extends over all unique atoms i of the system. The quantity fi denotes the orthogonal coordinates of atom i in A. T is the 3x3 matrix that converts orthogonal (A) coordinates into fractional coordinates F is its transpose. Bi, qi are respectively the atomic temperature factor and occupancy for atom i. The atomic form factors /j(/i) are typically approximated by an expression consisting of several Gaussians and a constant [19]... 

Some types of matrices that appear frequently are Hermitian matrices, for which A = A anti-Hermitian matrices, for which A = — A and unitary matrices, for which = U The MCSCF methods diseussed in most detail in this review will involve only operations of real matrices. In this case these matrix types reduce to symmetric matrices, for which A = A , antisymmetric matrices, for which A = — A, and orthogonal matrices, for which U = U. A particular type of orthogonal matrix is called a rotation matrix and satisfies the relation Det(R)= -I-1, where Det(R) is the usual definition of a determi-... 

The real transformation matrix U is actually a rotation matrix since Det(U) = Det(exp(K)) = exp(Tr(K)) = exp(0)= +1. The orbital phases are not important in the MCSCF method so that this loss of generality, compared to more general orthogonal transformations, is not significant. There are two representations of the K operator that are useful. The first results directly from Eq. (108) and is given as... 

An orthogonal transformation matrix U may be found to diagonalize the symmetric matrix C of Eq. (220). The matrix may be faetored, as U = Us, into the product of a rotation matrix U and a diagonal sign matrix s where = 1. The matrix U may further be written as U = exp(K) where K is antisymmetric as discussed in Section II. Eq. (18) shows Det(U) = exp(Tr(K)) = -I-1 as required for rotation matrices. An arbitrary two-electron wavefunction may then be written in the form... 

Suppose for example a trial wavefunction has the form of Eq. (226) and is written explicitly in terms of the n pair-expansion variables and the n(n — l)/2 variables in the matrix K. The theory of the orthogonal rotation group (or of the more general unitary group) may be invoked to factor the orthogonal matrix exp(K) according to... 

The orthogonal transformation which allows the connection between Yjm (9, ) and Yhn 9, 4> ) is in this case a Wigner s rotation matrix. For d greater than three the number of possible representations increases and we can have different hyperspherical harmonics. Transformations allowing the connection among them are orthogonal matrices, whose study is a current important topic of angular momentum theory. 

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