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Symmetries and Group Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Much like the orthogonal group is defined by relating the inverse of a matrix to its transpose, an nxn matrix V belongs to the unitary group U(n) if its Hermitian conjugate V†, i.e., the matrix obtained by transposing the matrix and taking the complex conjugate of each element, is its inverse () VV†=V†V=I.
All About Wave Equations
Published in Bahman Zohuri, Patrick J. McDaniel, Electrical Brain Stimulation for the Treatment of Neurological Disorders, 2019
Bahman Zohuri, Patrick J. McDaniel
Note that, in mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The group operation is that of matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n × n unitary matrices, which is itself a subgroup of the General Linear group GL(n, C). More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.
Consensus and coordination on groups SO(3) and S 3 over constant and state-dependent communication graphs
Published in Automatika, 2021
Aladin Crnkić, Milojica Jaćimović, Vladimir Jaćimović, Nevena Mijajlović
Consider a system of N agents whose states are described by points on manifold . Let be a special orthogonal group or a special unitary group . Suppose that agents communicate to each other through complete communication graph G. Based on the information received from his neighbours, each agent continuously adjusts his state. Following an analogy with linear consensus algorithms it is natural to consider the following minimization problem [2,3]: with respect to agents' states . The notion above stands for the transpose (or conjugate transpose) of a matrix X.
Convex feasibility problems on uniformly convex metric spaces
Published in Optimization Methods and Software, 2020
Byoung Jin Choi, Un Cig Ji, Yongdo Lim
Let be the special unitary group of degree 2. Put Then we can identify with by the map defined by Consider six elements in T as following: Then the points given by are corresponding to by the map Ψ, and then the closed convex sets and can be considered as closed convex subsets of generated by the subsets and , respectively. It is clear that the point is on the geodesic connecting and , and so (see Figure 1).
Development of Xi'an-CI package – applying the hole–particle symmetry in multi-reference electronic correlation calculations
Published in Molecular Physics, 2018
Bingbing Suo, Yibo Lei, Huixian Han, Yubin Wang
The matrix element of a spin-free Hamiltonian is written as Here, (p|h1|q) and (pq|h12|rs) are one- and two-electron molecular orbital integrals in the Mulliken notation. The are the generators of the unitary group. The a+pσ and apσ are creation and annihilation operators for an electron on the orbital p with spin σ. In GUGA, the CI space is spanned by the Gelfand states. Each Gelfand state is written as a three-column Paldus tableau [8,9].