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Math Help - Invariant Subspace Representation

  1. #1
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    Invariant Subspace Representation

    If S1, S2 are T-invariant subspaces of V, with the intersection of S1 and S2 being the zero vector, then find the form of the representation of T in a basis {alpha(1),...,alpha(n1), beta(1),...,beta(n2), v(1),...,v(n3)} and alpha vectors represent S1, beta represent S2, and v vectors represent V.

    I really don't know what "the representation of T" means. Is it talking about the fact that T can be written in block diagonal form?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by letitbemww View Post
    If S1, S2 are T-invariant subspaces of V, with the intersection of S1 and S2 being the zero vector, then find the form of the representation of T in a basis {alpha(1),...,alpha(n1), beta(1),...,beta(n2), v(1),...,v(n3)} and alpha vectors represent S1, beta represent S2, and v vectors represent V.

    I really don't know what "the representation of T" means. Is it talking about the fact that T can be written in block diagonal form?
    I think what they're saying is that if S_1\cap S_2=\{\bold{0}\} and \{\alpha_1,\cdots,\alpha_{n_1} is a basis for S_1, \{\beta_1,\cdots,\beta_{n_2}\} is a basis for S_2, and \{\gamma_1,\cdots,\gamma_{n_3}\} are vectors obtained by extending \{\alpha_1,\cdots,\alpha_{n_1},\beta_1,\cdots,\bet  a_{n_2}\} to a basis for V then if \mathcal{B} is the ordered basis (\alpha_1,\cdots,\alpha_{n_1},\beta_1,\cdots,\beta  _{n_2},\gamma_1,\cdots,\gamma_{n_3}) then [T]_\mathcal{B} can be written as D_1\oplus D_2\oplus M where D_1,D_2 are diagonal.
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  3. #3
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    Oh I think I get it up to where you have M. What is M? Just any nxn matrix?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by letitbemww View Post
    Oh I think I get it up to where you have M. What is M? Just any nxn matrix?
    No, I must have been half-asleep when I wrote this--what I said was wrong assuming I'm interpreting you correctly. Let D and D' be the matrices one gets when computing \left[T_{\mid S_1}\right]_{(\alpha_1,\cdots,\alpha_n)} and similarly for D'. Then, to get your matrix imagine taking the block diagonal matrix D\oplus D' then padding "zero rows" below them to get a (n_1+n_2+n_3)\times(n_1+n_2) matrix and then add on n_3 columns which can be anything to get a (n_1+n_2+n_3)\times(n_1+n_2+n_3) matrix. Does that make sense
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    Oh that makes a little more sense. I'm just wondering why it can be anything in the n3 columns. I thought it would be the direct sum of D, D', and [T|V]_(v_1,...,v_n3). How can it be anything in the n3 columns?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by letitbemww View Post
    Oh that makes a little more sense. I'm just wondering why it can be anything in the n3 columns. I thought it would be the direct sum of D, D', and [T|V]_(v_1,...,v_n3). How can it be anything in the n3 columns?
    For the first two collections of columns (those corresponding to the bases of S_1 and S_2) we know that T is invariant under those 'blocks' and so the first two collections of columns will look like a block diagonal matrix. We don't know anything about the third collection of columns (in terms of T) so they can be anything, yeah?
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  7. #7
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    Quote Originally Posted by Drexel28 View Post
    For the first two collections of columns (those corresponding to the bases of S_1 and S_2) we know that T is invariant under those 'blocks' and so the first two collections of columns will look like a block diagonal matrix. We don't know anything about the third collection of columns (in terms of T) so they can be anything, yeah?
    Oh that's right. I didn't think about that. Thank you!
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