= Solution
Gershgorin's theorem states that every complex <eigenvalue> of a square matrix lies in at least one row disc
$$
D_i=\{z\in\mathbb C:|z-a_{ii}|\le R_i\},\qquad R_i=\sum_{j\ne i}|a_{ij}|.
$$
Let $Av=\lambda v$ with $v\ne0$, and choose an index $i$ where $|v_i|$ is maximal. Its row equation gives
$$
|\lambda-a_{ii}|\,|v_i|=\left|\sum_{j\ne i}a_{ij}v_j\right|\le\sum_{j\ne i}|a_{ij}|\,|v_i|.
$$
Dividing by the positive $|v_i|$ proves $\lambda\in D_i$. This proof applies to complex eigenvectors even when the matrix is real.
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