Let $K,C,M$ be 3 $n \times n$ real regular matrices. Let $v_i, i = 1,...,k$ an orthonormal basis for a $k$-dimensional subspace of $\mathbb{C}^n$ and define $$V = [v_1, v_2, ...,v_k]$$ Let $F(\omega) = K + i \omega C- \omega^2 M$ and let $z(\omega)$ implicitly given by $$V^* F(\omega) V z(\omega) = V^* f$$ where $.^*$ is the conjugate transpose. How can I calculate the derivative of $z(\omega)$ in $0$ ? The problem is that I don't know how to deal with the fact that the matrix is only given implicitely.
Can someone help me?