Orthogonal transformation, definition

Orthogonal transformations preserve the lengths of vectors. If the same orthogonal transformation is applied to two vectors, the angle between them is preserved as well. Because of these restrictions, we can think of orthogonal transfomiations as rotations in a plane (although the formal definition is a little more complicated). 

Operator definitions, phase properties, 206-207 Optical phases, properties, 206-207 Orbital overlap mechanism, phase-change rule, chemical reactions, 450-453 Orthogonal transformation matrix ... 

By comparison with (Al), equation (A5) implies the definition of an orthogonal transformation matrix T(p)... 

According to this definition, a tensor of the first rank is simply a vector. As examples of second rank tensors within classical mechanics one might think of the inertia tensor 6 = (0y) describing the rotational motion of a rigid body, or the unity tensor 5 defined by Eq. (2.11). A tensor of the second rank can always be expressed as a matrix. Note, however, that not each matrix is a tensor. Any tensor is uniquely defined within one given inertial system IS, and its components may be transformed to another coordinate system IS. This transition to another coordinate system is described by orthogonal transformation matrices R, which are therefore not tensors at all but mediate the change of coordinates. The matrices R are not defined with respect to one specific IS, but relate two inertial systems IS and IS. ... 

A matrix product of the form A" HA is called a similarity transformation on H. If A is orthogonal, then AHA is a special kind of similarity transformation, called an orthogonal transformation. If A is unitary, then A HA is a unitary transformation on H. There is a physical interpretation for a similarity transformation, which will be discussed in a later chapter. For the present, we are concerned only with the mathematical definition of such a transformation. The important feature is that the eigenvalues, or latent roots, of H are preserved in such a transformation (see Problem 9-5). 

The covariance matrix is factored using the diagonal matrix A and the eigenvector matrix U as U UT. Since 5 is symmetric and positive-definite, the eigenvalues are positive and the eigenvectors orthogonal. The inverse S 1 of S can be expanded as UA 1UT and the transformation... 

Thereby the matrix 11(F) denotes a K-dimensional permutation matrix and F (F) a 3 by 3 orthogonal matrix. The form of this representation follows from the fact that each isometric transformation maps the NC Xk, Zk, Mk onto a NC which by definition has the same set of distances, i.e. is isometric to NC Xk, Zk, Mk. Expressed alternatively, the nuclear configurations NC Xk( ), Zk, Mk and NC Xk(F 1 ( )), Zk, Mk are properly or improperly congruent up to permutations of nuclei with equal charge and mass for any F G ( ). The set of matrices Eq. (2.12) forms a representation of J d) by linear transformations and will furtheron be denoted by... 

If the coordinate transformation R is chosen so that D = lrAI, where A is a diagonal matrix, then (recalling the fact that the eigenvectors of a positive definite matrix are orthogonal) A = 1DK7 and the diffusion equation is... 

In PCA the input descriptors are transformed into orthogonal principal components with a small number of principal components usually being sufficient to represent most of the variation in the original data. PCA can be very useful when the descriptors are correlated since by definition the new set of descriptors will be uncorrelated however, a disadvantage of the approach is that it can be difficult to interpret the resulting models since each new descriptor represents a linear combination of the original descriptors. 

The purpose of such a device consists in changing the orientation of the polarization plane of a beam by 90°. That means the initial Stokes vector 1,1,0,0 of a horizontally polarized beam becomes 1,-1,0,0 after passing through the retarder. Retarders are most often birefringent crystals of definite thickness. If the fast and slow axes of such a crystal orthogonal to each other are crossed at 45° with respect to the polarization plane, the retarder rotates the latter by 90°. The Stokes-Mueller transformation corresponding to this experiment should be ... 

The definition of the invariant quantities, the 6-1 and the 9-1 S5nnbols may be inferred from a consideration of the orthogonal recoupling transformation between the different coupling schemes of irreducible products of the same degree ... 

The integral form of this orthogonality condition is identical to the definition of the integral transform (11.15) if we replace y with KJix) and see... 

Nonorthogonal vectors, qj,..., q are transformed into r orthogonal vectors Pj, Pj,..., Pr- The remaining vectors p are preserved as they are orthogonal to each other, but also orthogonal with respect to the new vectors, since they do not appear in the definition q. This transformation can be carried out by various different techniques, such as the Gram-Schmidt transformation (Golub and Van Loan, 1983). 

If the molecules has some symmetry the solution of the secular determinant can be simplified by the use of symmetry coordinates. Symmetry coordinates are linear combinations of the internal coordinates. After a little experience one can frequently select symmetry coordinates intuitively by taking combinations of symmetry related internal coordinates. The symmetry coordinates picked out are not necessarily unique but must have definite properties. They must be normal, orthogonal, and they must transform properly as will be described below. A symmetry coordinate has the form... 

One definition of factor analysis is given by Malinowsky in [28] "Factor analysis is a multivariate technique for reducing matrices of data to their lowest dimensionality by the use of orthogonal factor space and transformations that yield predictions and/or recognizable factor . 

Fig. 7.9. Notions used in the definition of the Cauchy strain tensor The material point at r in the deformed body with its neighborhood shifts on unloading to the position r. The orthogonal infinitesimal distance vectors dri and drs in the deformed state transform into the oblique pair of distance vectors dri and dr. Orthogonality is preserved for the distance vectors dra, drc oriented along the principal axes...

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