= Solution
The <Gershgorin circle theorem> states that every complex <eigenvalue> of a square <matrix> $A$ lies in at least one closed disc $D_i=\{z:|z-a_{ii}|\le\sum_{j\ne i}|a_{ij}|\}$. To prove it, take a nonzero <eigenvector> $v$ for $\lambda$ and choose $i$ with $|v_i|$ maximal. The $i$th row of $Av=\lambda v$ gives
$$
|\lambda-a_{ii}|\,|v_i|
=\left|\sum_{j\ne i}a_{ij}v_j\right|
\le\sum_{j\ne i}|a_{ij}|\,|v_j|
\le\left(\sum_{j\ne i}|a_{ij}|\right)|v_i|.
$$
Division by $|v_i|>0$ proves the claim. The proof applies to real and complex <matrices> alike.
Back to article page
