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Functional Equations on Affine Groups
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
In the previous examples V was a real vector space. However, most of these linear groups have their counterpart in the complex case. Let now V be a complex vector space. The unitary group U(n) is the group of all linear transformations k in GL(V) whose matrix is unitary: [k]*[k]=In, where [k]* is the adjoint of [k]: the transpose of its complex conjugate. The special unitary group SU(n) is the set of those unitary transformations with determinant +1, i.e. SU(n)=SL(V)∩U(n).
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.
Symmetries and Group Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
In the case of O(n), we defined the subgroup SO(n) as the matrices in O(n) that have determinant one. In complete analogy, the special unitary group SU(n) is defined as the subgroup of U(n) that consists of unitary matrices of determinant one.
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).
An optical channel modeling of a single mode fiber
Published in Journal of Modern Optics, 2018
Neda Nabavi, Peng Liu, Trevor James Hall
In a coherent detection scheme, the coherent receiver must be phase diverse to compensate for fluctuations in the overall phase (47). It follows from that one is at liberty to set when the state of polarization only of interest, i.e. may be chosen as a member of the special unitary group in two dimensions SU(2). Every member of SU(2) has the Caley–Klein representation (48):