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New PDF release: (A, B)-Invariant Polyhedral Sets of Linear Discrete-Time

By Dorea C. E.

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Proof. Assume that M commutes with all the π(g). We want to show that (π, V ) irreducible implies M = λ1. Since we are working over the field C (this doesn’t work for R), we can always solve the eigenvalue equation det(M − λ1) = 0 to find the eigenvalues λ of M . The eigenspaces Vλ = {v ∈ V : M v = λv} are non-zero vector subspaces of V and can also be described as ker(M − λ1), the kernel of the operator M −λ1. Since this operator and all the π(g) commute, we have v ∈ ker(M − λ1) =⇒ π(g)v ∈ ker(M − λ1) so ker(M − λ1) ⊂ V is a representation of G.

3! 1) As θ goes from θ = 0 to θ = 2π, this exponential traces out a circle in the space of unitary 2 by 2 matrices, starting and ending at the unit matrix. This circle is a group, isomorphic to U (1). So, we have found three different U (1) 28 subgroups inside the unitary 2 by 2 matrices, but only one of them (the case j = 3) will act diagonally on H, with the U (1) representation determined by Q= 1 0 0 −1 For the other two cases j = 1 and j = 2, by a change of basis either one could be put in the same diagonal form, but doing this for one value of j makes the other two no longer diagonal.

One motivation for the notation Q is that this is the conventional physics notation for electric charge, and this is one of the places where a U (1) group occurs in physics. Examples of U (1) groups acting on physical systems include: • Quantum particles can be described by a complex-valued “wavefunction” (see chapter 10), and U (1) acts on such wavefunctions by pointwise phase transformations of the value of the function. This phenomenon can be used to understand how particles interact with electromagnetic fields, and in this case the physical interpretation of the eigenvalue of the Q operator will be the electric charge of the state.

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(A, B)-Invariant Polyhedral Sets of Linear Discrete-Time Systems by Dorea C. E.


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