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    I 2 and 3 dimensional invariant subspaces of R4

    I am looking at the representation of D4 in ℝ4 consisting of the eight 4 x 4 matrices acting on the 4 vertices of the square a ≡ 1, b ≡ 2, c ≡ 3 and d ≡ 4. I have proven that the 1-dimensional subspace of D4 in ℝ2 has no proper invariant subspaces and therefore is reducible. I did this in 2...
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    A Block Diagonalization - Representation Theory

    This will take me a little time to digest, but in the meantime I wanted to thank everyone for your responses.
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    A Block Diagonalization - Representation Theory

    OK. Thanks. It seems that this involves numerical analysis and is best solved using matlab etc. Is that a fair assessment?
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    A Block Diagonalization - Representation Theory

    How does one go about finding a matrix, U, such that U-1D(g)U produces a block diagonal matrix for all g in G? For example, I am trying to figure out how the matrix (7) on page 4 of this document is obtained.
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    I Block Diagonal Matrix and Similarity Transformation

    I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/Lecture_3.pdf How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S...
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    I Rings, Modules and the Lie Bracket

    Thank you. I think this helps a lot. I realize now that modules and rings were somewhat of a red herring. However, I learned something that I didn't know before. Again, thanks for your patience and time.
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    I Rings, Modules and the Lie Bracket

    Maybe abstract is the wrong term to use. The tangent space has the structure of the Lie algebra which seems more elaborate than just vector addition and scalar multiplication. That is the source of my confusion. I understand the relevance of the LIVF but I can't connect the dots between...
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    I Rings, Modules and the Lie Bracket

    OK. I recognize the fact there is some 'backwards and forwards' between the 2 disciplines. I am a retired EE so am somewhat impartial. So the bottom line is that In the first case, forget about rings and modules since g, h ∈ G is not compatible with g ∈ G and h ∈ M. This makes perfect...
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    I Rings, Modules and the Lie Bracket

    Thank you for your reply (and patience!). My use of matrix ring may be incorrect. I wanted it to mean that the elements of the ring are matrices and these matrices act on vectors that are elements of the module. I don't know if that would effect your answer. Generic means a space equipped...
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    I Rings, Modules and the Lie Bracket

    OK. I forgot about the requirement for an Abelian group structure - a matrix ring operating on module elements that are vectors would be fine because the vectors form an Abelian group. I think I may also be confusing 'generic' vector spaces associated with abstract algebra with tangent...
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    I Rings, Modules and the Lie Bracket

    I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups. Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e, Lgh = gh where g,h ∈ G Which corresponds to a translation by g. Is this an example of a module over a ring...
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    B Tensor Product, Basis Vectors and Tensor Components

    I am trying to figure how to get 1. from 2. and vice versa where the e's are bases for the vector space and θ's are bases for the dual vector space. 1. T = Tμνσρ(eμ ⊗ eν ⊗ θσ ⊗ θρ) 2. Tμνσρ = T(θμ,θν,eσ,eρ) My attempt is as follows: 2. into 1. gives T = T(θμ,θν,eσ,eρ)(eμ ⊗ eν ⊗ θσ ⊗ θρ)...
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