Optimization problem of the ratio of two quadric forms $f(x) = \frac{x^H A x}{x^H B x}$

Question

Let $A, B \in \mathbb{C}^{n\times n}$ be Hermitian matrices ($A^H = A$, $B^H = B$). $x \in \mathbb{C}^{n \times 1}$ is the decision variable, then optimization problem is \begin{equation} \max\colon f\left( x \right) = \frac{x^H A x}{x^H B x}, \quad \text{s.t. $\lVert x \rVert^2 = 1$.} \end{equation}

Questions:

1. How to compute the maximum value of $f(x)$ and corresponding $x$?
2. How to solve the question 1 when the problem is $\min\colon f(x)$?

Answer 1

Converted from a comment:

The quotient $x^H\!Ax / x^H\!Bx$ is scaling invariant so you can ignore the constraint $\|x\| = 1$ as long as $x \neq 0$. We can probably assume that $B$ is definite, else $x^H \! B x$ can vanish and we can usually make the quotient arbitrarily negative or positive by perturbing $x$. If $B$ is definite, choose an orthonormal basis for the Hermitian form $x^H \! Bx$ and change to that basis so the denominator becomes $\pm \|x\|^2$. Now you have a standard Rayleigh quotient and you just need the largest and smallest eigenvalues of the Hermitian matrix in the numerator.