I am trying to solve the following problem:
For given complex Hermitian symmetric matrices $\hspace{0.2cm} A_1, A_2, \dots ,A_M \in \mathbb{C}^{N \times N} \hspace{0.2cm}$, determine whether a vector $\hspace{0.2cm} x\in \mathbb{C}^{N \times 1}\hspace{0.2cm}$ with $||x||_2=1$ exists or not such that $\quad x^HA_kx \geq 0 \quad \forall k \in \{1,2,\dots,M\}$. If such an $\hspace{0.1cm}x\hspace{0.1cm}$ exists, how to find it? Particularly in my case, $M=N$.
I don't know if this is a very difficult problem or not.
I was originally trying to solve the following problem:
For given positive definite complex Hermitian symmetric matrices $\hspace{0.2cm} W_1, W_2, \dots ,W_M \in \mathbb{C}^{N \times N}$, find $x$ with $||x||_2=1$ such that $\prod_{k=1}^{M}x^HW_kx$ is maximized. I found some papers that provide sufficient conditions (based on the condition number of $W_i^{\frac{-1}{2}}W_jW_i^{\frac{-1}{2}}$, see paper "A CONDITION FOR CONVEXITY OF A PRODUCT OF POSITIVE DEFINITE QUADRATIC FORMS") so that the product is convex, however those conditions are not satisfied in my case, so the product in my case is concave.
I know that $\sum_{k=1}^{M}x^HW_kx=x^H\left(\sum_{k=1}^{M}W_k\right)x \leq \lambda$ where $\lambda$ is the largest eigenvalue of the matrix $\sum_{k=1}^{M}W_k$. So, summation of the terms in the product is upper bounded. Therefore, I try to make each term as close as to each other so that product is maximized or try to keep the smallest one greater than some value (although this heuristic method does not guarantee to give the best possible $x$, it might still give a nice result). To do that, I look for a $\gamma$ such that $x^HW_kx \geq \gamma$ $\forall k$. Then this trial reduces to $x^H(W_k-\gamma I)x \geq 0$ $\forall k$. I substituted $(W_k-\gamma I)=A_k$ and ask the question here. If the question I ask at the very top can be solved, then I will be able to find largest $\gamma$ value and corresponding $x$ vector, and finally the product will be large enough perhaps.