Suppose I have a vector valued function on $[0,1] $ where $ \mathbf{x}:[0,1] \to \mathbb{C}^k $ given by $\mathbf x(\omega) = [x_1(\omega), \dotsc x_k(\omega) ]$. Then suppose I have a symmetric postive semi-definite matrix valued function defined similarly. That is $A: [0,1] \to \operatorname{Mat}_{k \times k}(\mathbb{C}) $ with $A = \{A_{mn}(\omega) \}_{1 \leq m,n \leq k} $ and for all $\omega $, $A(\omega)$ is symmetric and non-negative definite.
I want to know if there's anything we can say about
$$ \int_0^1 \overline{\mathbf{x}(\omega)}^TA(\omega)\mathbf{x}(\omega) \ d\omega $$
in terms of the eigenvalues of $\{A(\omega)\}_{\omega \in \mathbb{[0,1]}} $ and $$||\textbf{x}(\omega) ||_{L^2[0,1]} := \int_0^1 ||x(\omega)||_{\ell^2}^2 \ \ d \omega $$
where $\overline{\textbf{x}(\omega)}$ is the complex conjugate of $\textbf{x}({\omega}) $. I expect something like $$\lambda_{min}||x(\omega)||_{L^2[0,1]}^2 \leq \int_0^1 \overline{\mathbf{x}(\omega)}^TA(\omega)\mathbf{x}(\omega) \ d\omega\leq \lambda_{max}||x(\omega)||_{L^2[0,1]}^2 $$
I may need more conditions on $A(\omega)$, but I am curious if such results exist.