General Rayleigh quotient

Question

I have a question that you might help. Let's suppose that we want to maximize a Rayleigh quotient $f(\bf{x})= \frac{\bf{x}^H\bf{p}\bf{p}^H\bf{x}}{\bf{x}^H\bf{Q}\bf{x}} $. We know that it gets its maximum at $\bf{x}= \frac{\bf{Q}^{-1}\bf{p}}{\|\bf{Q}^{-1}\bf{p}\|}$. Now, my question is how we can solve it if we have a constraint which should satisfy $\bf{x}^{H}\bf{U}\bf{q}=0$. Can we derive any close-form for it? $\begin{align} \max_{\bf{x}}& \hspace{1em} f(\bf{x})= \frac{\bf{x}^H\bf{p}\bf{p}^H\bf{x}}{\bf{x}^H\bf{Q}\bf{x}}, s.t \hspace{1em} \bf{x}^{H}\bf{U}\bf{q}=0 \end{align}$

Answer 1

You only need to perform a change of basis. Presumably $\mathbf U\mathbf q\ne0$. Pick a unitary matrix $\mathbf V$ whose last column is $\mathbf U\mathbf q/\|\mathbf U\mathbf q\|_2$ and let \begin{aligned} \mathbf V^H\mathbf x=\pmatrix{\mathbf x_1\\ y}, \ \mathbf V^H\mathbf p=\pmatrix{\mathbf p_1\\ \ast}, \ \mathbf V^H\mathbf Q\mathbf V=\pmatrix{\mathbf Q_1&\#\\ \#&\ast}, \end{aligned} where $y$ is a scalar, the asterisks denote unspecified scalars and the symbol $\#$ denotes a row or column vector. The constraint $\mathbf x^H\mathbf U\mathbf q$ is then equivalent to $y=0$. Subject to this constraint, we have $$ f(\mathbf x) =\frac{\mathbf x^H\mathbf p\mathbf p^H\mathbf x}{\mathbf x^H\mathbf Q\mathbf x} =\frac{(\mathbf x^H\mathbf V)(\mathbf V^H\mathbf p)(\mathbf p^H\mathbf V)(\mathbf V^H\mathbf x)}{(\mathbf x^H\mathbf V)(\mathbf V^H\mathbf Q\mathbf V)(\mathbf V^H\mathbf x)} =\frac{\mathbf x_1^H\mathbf p_1\mathbf p_1^H\mathbf x_1}{\mathbf x_1^H\mathbf Q_1\mathbf x_1}. $$ So, if you can find a vector $\mathbf x_1$ that maximises the RHS above, then $\mathbf x=\mathbf V\pmatrix{\mathbf x_1\\ 0}$ will be a solution to your original problem.