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Linear Algebra for Coupled Oscillators

  1. Jan 9, 2013 #1
    Hi guys, first post!

    I'm doing some side work on coupled oscillating systems, and I've almost found a procedure to get the analytic solution (an n-dimentional vector function of time) to a mass-spring system in which n masses are arraigned in a line and separated by springs, but I need help solving this equation for D. I'm asking a pure math question here, you can ignore all the above if you like.

    DNF=SDN
    D and F are diagonal.
    S is a sparse symmetric matrix.
    I think that N is invertible.
    All the matrices are square nxn.

    _____

    Solving D on either side seems pointless because you can't isolate it (as far as I can see).

    DNF=SDN ---> D=SDNF-1N-1
    DNF=SDN ---> S-1DNFN-1=D

    Can I simplify NF-1N-1 in the first line or NFN-1 in the second, using the fact that F and its inverse are diagonal? Obviously I can't. But can you?
    _____

    I tried taking the transpose of both sides of the original equation and using the fact that the transpose of a symmetric matrix is itself:

    DNF=SDN
    (DNF)T=(SDN)T
    FTNTDT=NTDTST
    FNTD=NTDS

    So now I have two equations, the second derived from the first and making use of what I see to be everything we know in general about these matrices:

    DNF=SDN
    FNTD=NTDS
    _____

    Any suggestions on where to go from here?

    Thanks,
    Jack
     
  2. jcsd
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