The Generalised Eigenvalue Problem

A common generalisation of the simple eigenvalue problem involves 2 matrices

$$\bss{A}\bi{x} = \alpha\bss{B}\bi{x}.$$ (3.29)

This can easily be transformed into a simple eigenvalue problem by multiplying both sides by the inverse of either $\bss{A}$ or $\bss{B}$. This has the disadvantage however that if both matrices are Hermitian $\bss{B}^{-1}\bss{A}$ is not, and the advantages of the symmetry are lost, together, possibly, with some important physics. There is actually a more efficient way of handling the transformation. Using Cholesky factorisation an $\bss{L}\bss{U}$ decomposition of a positive definite matrix can be carried out such that

$$\bss{B} = \bss{L}\bss{L}^\dagger$$ (3.30)

which can be interpreted as a sort of square root of $\bss{B}$. Using this we can transform the problem into the form

$$\left[\bss{L}^{-1}\bss{A}\left(\bss{L}^\dagger\right)^{-1}\right] \left[\bss{L}^\dagger \bi{x}\right] = \alpha \left[\bss{L}^\dagger \bi{x}\right]$$ (3.31)

$$\bss{A}'\bi{y} = \alpha\bi{y}.$$ (3.32)

Most libraries contain routines for solving the generalised eigenvalue problem for Hermitian and Real Symmetric matrices using Cholesky Factorisation followed by a standard routine. Problem 6 contains a simple and informative example.