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Math Help - Spectral radius of sum of commuting matrices

  1. #1
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    Spectral radius of sum of commuting matrices

    If two matrices A and B commute, when is it true that rho(A+B) = rho(A)+rho(B), i.e. that the spectral radius of A+B is equal to the sum of the two spectral radii or A and B?
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    MHF Contributor Bruno J.'s Avatar
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    What if B=-A?
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    Thanks, with B = -A I guess it holds trivially, is there any other set of conditions that ensures this?
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    MHF Contributor Bruno J.'s Avatar
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    Why does it hold? The spectral radius of the zero matrix is zero; so if your formula held, every matrix would have a zero spectral radius.
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  5. #5
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    shoot! of course

    but is it true at least that if A and B commute then rho(A+B)<=rho(A)+rho(B)? I think yes. My question is really, when does this hold with equality.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by chrysi View Post
    shoot! of course

    but is it true at least that if A and B commute then rho(A+B)<=rho(A)+rho(B)? I think yes. My question is really, when does this hold with equality.
    Sorry for reading your question too fast.
    It might be true!
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