# Hermitian General Eigenvalue Problem

1. Jul 28, 2013

### HasuChObe

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
$${\bf Ax}_n=\lambda_n{\bf Bx}_n \\ {\bf v}=\sum_nc_n{\bf x}_n$$

when is the following true:
$$c_n=\frac{{\bf x}_n^T{\bf Bv}}{{\bf x}_n^T{\bf Bx}_n}$$

and why. In my particular problem, I believe ${\bf A}$ is singular and hermitian. I've also read that ${\bf B}$ 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
$$\frac{{\bf x}_i^T{\bf Bx}_j}{{\bf x}_i^T{\bf Bx}_i}=\delta_{i,j}$$

Last edited: Jul 28, 2013