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Computational Vector Geometry
Published in Hannah Robbins, Functional Linear Algebra, 2021
One reason to use an orthogonal basis is that it makes finding coordinate vectors much faster. If you recall from 3.1, we can find the coordinate vector of v→ with respect to a basis B={b→1,…,b→n} by solving the vector equation v→=x1b→1+⋯+xnb→n
Inner Product Spaces
Published in Lina Oliveira, Linear Algebra, 2022
A key feature of an orthogonal basis is how simple it is to calculate the coordinates of any vector relative to said basis. Let x be a vector in the inner product space V and let B=(b1,…,bn) be an ordered orthogonal basis of V. Suppose that the coordinate vector of x relative to B is xB=(α1,…,αn),
Matrices and Linear Algebra
Published in William S. Levine, The Control Handbook: Control System Fundamentals, 2017
Let V be an inner product space, a vector space with an inner product and compatible norm. A set of mutually orthogonal vectors is known as an orthogonal set, and a basis consisting of mutually orthogonal vectors is known as an orthogonal basis. An orthogonal basis consisting of vectors whose norms are all one (i.e. consisting of vectors having unit length) is called an orthonormal basis.
A hybrid algorithm of orthogonal perturbation method and convex optimization for beamforming of sparse antenna array
Published in Electromagnetics, 2020
Lei Liang, Can Jin, Hailin Li, Jialing Liu, Yachao Jiang, Jianjiang Zhou
Based on the analysis of the advantages of OPM and CVX algorithm, a hybrid algorithm of OPM-CVX is proposed. On the premise of proper initial array layout, the OPM is used to obtain the optimal array position through the iteration process, and the CVX optimization algorithm is used to obtain the excitation coefficients of sparse array. The advantages of the two algorithms ensure the optimization efficiency of the element position and the global optimal result of the excitation coefficient. The following are the detailed descriptions of the hybrid OPM-CVX algorithm. Set the element number and aperture of the specific array model and initialize the array layout.Set the desired pattern.Optimize the beamforming process by OPM-CVX. Construct orthogonal bases by Gram-Schmidt orthogonal process.Update perturbation position parameters.Optimize the excitation coefficients of array elements by CVX optimization model (14).Calculate the actual electric field intensity using the updated position and excitation coefficients.Increase the loop counter. If the position is optimal, execute step (4), otherwise execute step (3).Optimize the excitation coefficients of array elements by CVX optimization model (15).Display the best results of pattern synthesis.End.
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 , .