I've found this inequality being used a lot in papers but have been unable to prove it for myself.
Consider a matrix polynomial $P(z)=A_{m}z^{m}+...+A_0$ with complex matrix coefficients. Then, for each eigenvalue $\lambda$ of $P(z)$ and some unit eigenvector $x\in \mathbb{C}^m$ associated with $\lambda$, we have $$||P(\lambda)x|| \geqslant ||A_{m}\lambda^{m}x||-||A_{m-1}\lambda^{m-1}x+...+A_0x||$$