Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Approximate diagonalisation of (3,3) hermitian matrix

  1. Feb 3, 2014 #1
    Hi,

    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?

    Thanks,
    Susy
     
  2. jcsd
  3. Feb 3, 2014 #2

    AlephZero

    User Avatar
    Science Advisor
    Homework Helper

    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).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Approximate diagonalisation of (3,3) hermitian matrix
  1. A3 is a 3 x 3 matrix (Replies: 3)

  2. 3 by 3 matrix puzzle? (Replies: 3)

  3. Matrix diagonalisation (Replies: 2)

  4. Diagonalising a matrix (Replies: 4)

Loading...