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Planning Method of Scanning Trajectory on Free-Form Surface
Published in Chunguang Xu, Robotic Nondestructive Testing Technology, 2022
In the above case, attitude is represented by a 3 × 3 rotation matrix, which is a special unit matrix with orthogonal columns. Therefore, the determinant of any rotation matrix is always +1 and is called standard orthogonal matrix. Since rotation matrix can be considered as both an operator and an attitude descriptor, undoubtedly it has different representations in different applications. Rotation matrix can be used as an operator. When multiplied by a vector, it plays the role of rotation operator. However, it is still a little complicated to determine the attitude with a rotation matrix containing nine elements, especially when determining the position and attitude changes of a large number of data points. Next, the coordinate-system attitude rotation method using three numbers to represent the position and attitude changes of a manipulator will be introduced:
Polarized Electromagnetic Waves
Published in José J. Gil, Razvigor Ossikovski, Polarized Light and the Mueller Matrix Approach, 2022
José J. Gil, Razvigor Ossikovski
At this point it is important to recall that, as seen in Section 1.6, unitary transformations play a key role in the study of coherence of electromagnetic fields in both space–time and space–frequency domains where the focus is on the correlations of the field variables (determined by the invariant degrees of coherence and by the degree of polarization) rather than in the particular states of polarization of the interfering waves. Nevertheless, when the focus is on polarization, two kinds of unitary transformations should be distinguished, namely orthogonal and nonorthogonal. Orthogonal transformations (hereafter rotation transformations) are represented by real valued unitary matrices (called orthogonal matrices) and can be physically realized by means of rotations of the Cartesian reference axes about the direction of propagation, so that they exclusively affect the azimuth, while the shape of the polarization ellipse is preserved. Nonorthogonal unitary matrices have necessarily off-diagonal elements with nonzero imaginary parts (i.e., cannot be realized through rotations of the laboratory axes) and necessarily produce changes in the ellipticity, so that the shape of the polarization ellipse undergoes the corresponding modification.
Introduction and Background
Published in Ossama Abdelkhalik, Algorithms for Variable-Size Optimization, 2021
Two vectors are said to be orthogonal if they are perpendicular to each other; that is is the scalar product of the two vectors is zero. The vectors {x→1,x→2,⋯,x→m} are said to be mutually orthogonal if every pair of vectors is orthogonal. A set of vectors is said to be orthonormal if every vector is of unit magnitude and the set of vectors are mutually orthogonal. A set of vectors that are orthogonal are also linearly independent.
An integrated solution for reducing ill-conditioning and testing the results in non-linear 3D similarity transformations
Published in Inverse Problems in Science and Engineering, 2018
The determinant of a square matrix also is a good criterion for detecting singularity of the matrix. At the same time, it is used as a supplemental tool to search the consistency. Sometimes the condition numbers given above is not enough to determine the consistency. If the determinant goes to zero (), a singularity problem will occur in the solution. In contrast to this, the determinant runs into the infinity (), the rounding errors can arise in the solution. The optimum value of a determinant of a square matrix is also equal to one (±1) (it is the orthogonal property for a matrix). The most optimum solution for a liner form is accomplished if and only if .
Cascade neural network algorithm with analytical connection weights determination for modelling operations and energy applications
Published in International Journal of Production Research, 2020
Zhengxu Wang, Waqar Ahmed Khan, Hoi-Lam Ma, Xin Wen
Suppose symmetric matrix has two different eigenvalues and corresponding to eigenvectors and in matrix respectively. Two vectors can be considered orthogonal if their inner product is zero, such as: or . where is the transpose of .
A theoretical investigation of time-dependent Kohn–Sham equations: new proofs
Published in Applicable Analysis, 2021
G. Ciaramella, M. Sprengel, A. Borzi
Let us recall some facts and existing results that we will use in this work. There exists an orthogonal basis for which is orthonormal in . Since , we can choose this basis to be , where are eigenfunctions of the Laplace operator. This follows from [22, Theorem 1 in 6.5.1 and Theorem 4 in 6.3.2] and [23, Theorem 2.5.1.1]. Throughout this paper, is used to denote this basis.For any integer m>0 and some coefficients , the functions and vanish on .For any , we can define , for and . Then the inequalities , and follow by Parseval–Plancherel's theorem and the orthogonality properties of .Consider the extension operator . Since , Theorem 4.32 in Section IV of [20] guarantees that is a continuous operator from to and from to for .Consider the Hartree potential and define . It follows from [12, Lemma 5] that there exist positive constants such that Consider the space Z. The norms and are equivalent, see, e.g. [24, Theorem 2.31]. We denote by the positive equivalence constant such that , .