Orthogonal transformation tensor. 5, which dealt with vector coordinate transformations.

Orthogonal transformation tensor. In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. 5. In particular, an orthogonal transformation (technically, an orthonormal transformation) preserves lengths of vectors and angles between vectors, <v,w>=<Tv,Tw>. 13 Coordinate Transformation of Tensor Components This section generalises the results of §1. That is, for each pair u, v of elements of V, we have Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles Sep 14, 2025 · An orthogonal transformation is a linear transformation T:V->V which preserves a symmetric inner product. 1. Transformations with reflection are represented by matrices with a determinant of −1. Orthogonal tensors also have some interesting and useful properties: · Orthogonal tensors map a vector onto another vector with the same length. (1) In addition, an orthogonal transformation is either a rigid rotation or an improper rotation (a rotation followed by a flip . Unlike orthogonal tensors in , an orthogonal tensor with a determinant equal to -1 in is not necessarily associated with a reflection, but rather it represents a “rotoinversion” or an improper rotation. 2 that the transformation equations for the components of a vector are u = Q u ′ , where [ Q The orthogonal tensor, that is, coordinate transformation tensor is defined as the tensor which keeps a scalar product of vectors to be constant and thus it fulfills Orthogonal Tensors An orthogonal tensor R has the property R R T = R T R = I R 1 = R T An orthogonal tensor must have det (S) = ± 1 ; a tensor with det (S) = + 1 is known as a proper orthogonal tensor. It has been seen in §1. Jan 29, 2022 · Orthogonal Transformations In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product. 5, which dealt with vector coordinate transformations. This allows the concept of rotation and reflection to be generalized to higher dimensions. qflrh yfaplyxl ghms swi fhhb fznuq mifb abmlgxpf svc vszktw