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Approximate diagonalisation of (3,3) hermitian matrix

  1. Feb 3, 2014 #1

    I have a 3 by 3 hermitian matrix K that I need to diagonalise. More accurately, I am searching for a unitary matrix S such that [tex]S^{\dagger} K S[/tex] is diagonal.

    The problem is that K is very complicated and the expression for S in mathematica takes up quiiiiiet a lot of space.

    Is it possible to find approximate expressions for the entries of S in terms of entries in K? What I am looking for is something like (as an example) [tex]S_{ij}=K_{ij} + K_{ji} bla bla[/tex]
    Or is it maybe possible to decompose S into matrices which I can then approximate by such expressions (they can, of course, be more complicated than the example given above).

    Since the entries in K are given as a power series of some small parameter ε [tex]K_{ij}=a_0 + a_1 \epsilon + ...[/tex] and I am only interested in the lowest non-vanishing order anyway, it would be nice to have expressions for the entries in S in terms of the entries in K. Then, I could easily evaluate the entries in S to lowest non-vanishing order in ε (so I am only interested in an approximate S anyway).

    Does anybody know of such a method?

  2. jcsd
  3. Feb 3, 2014 #2


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    If you can write ##K = K_0 + \epsilon K_\epsilon##, can you diagonalize ##K_0## and ##K_\epsilon## simultaneously? This is routinely done numerically, as a generalized eigenproblem, but I don't know about doing it symbolically.
  4. Feb 3, 2014 #3
    Yes, you should be able to diagonalize both simultaneously. In all cases I can think of, $K_0$ will already be a diagonal matrix and $K_{\epsilon}$ is a correction ($K_{\epsilon}$ can in principle be an arbitrary hermitian matrix).
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