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Hermitian General Eigenvalue Problem

  1. Jul 28, 2013 #1
    Hello~ I've been trying to understand where this orthogonality comes from but I haven't determined a solution. I was wondering if someone could shed some light on this problem. Basically, given
    [tex]
    {\bf Ax}_n=\lambda_n{\bf Bx}_n \\
    {\bf v}=\sum_nc_n{\bf x}_n
    [/tex]

    when is the following true:
    [tex]
    c_n=\frac{{\bf x}_n^T{\bf Bv}}{{\bf x}_n^T{\bf Bx}_n}
    [/tex]

    and why. In my particular problem, I believe [itex]{\bf A}[/itex] is singular and hermitian. I've also read that [itex]{\bf B}[/itex] may have to be positive definite for this orthogonality to occur. I'd be happy if both singular and non-singular cases are addressed.

    Thanks :)

    Edit:
    The orthogonality is
    [tex]
    \frac{{\bf x}_i^T{\bf Bx}_j}{{\bf x}_i^T{\bf Bx}_i}=\delta_{i,j}
    [/tex]
     
    Last edited: Jul 28, 2013
  2. jcsd
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