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Diagonalising an n*n matrix analytically

  1. Jun 14, 2015 #1
    Hi everyone
    I am trying to diagonalise a (2n+1)x(2n+1) matrix which has diagonal terms A_ll = (-n+l)^2 and other non vanishing terms are A_l(l+1) = A_(l+1)l = constant.
    Is there any way I can solve it for general n without having to use any numerical methods.
    I remember once a professor diagonalised such a matrix for a fixed value of n using some tricks, but I can't remember how he did that. Can anyone help me out here?

    Any help is appreciated. Cheers!
  2. jcsd
  3. Jun 14, 2015 #2


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    Staff: Mentor

    The Gauß algorithm should need about 4n steps (+- a few), and you can do it analytically and see if the result has some reasonable expression (but even if it does not, you get an analytic result).
  4. Jun 15, 2015 #3
    Hi ...
    I have managed to find an iterative expression to solve for the eigenvalues. But without selecting a fixed value of n I can't do anything with it. Any suggestions how I can go ahead with it.
    Thanks mfb
  5. Jun 15, 2015 #4


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    Finding the eigenvalues for a matrix is equivalent to solving the n degree eigenvalue equation. There is NO algebraic method for exactly solving a general polynomial of degree 5 or higher.
  6. Jun 16, 2015 #5
    Are you assuming that ##A_{j,j+1}= A_{j+1,j} = a ## for all ##j##? (i.e.> that ##a## is the same for all ##j##). If so, then the case when ##a=1## and all other entries are ##0## is the case of the so-called free Jacobi matrix. Its spectrum is computed in terms of Chebyshev polynomials. Your case then can be obtained by a simple affine transformation.

    Even if your ##a##s are different, look up Jacobi matrices and orthogonal polynomials.
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