Gershgorin Circle Theorem

Theorem
Let $n$ be a positive integer.

Let $A = \sqbrk {a_{i j} }$ be a complex square matrix of order $n$.

Let $\lambda$ be an eigenvalue of $A$.

Then there exists $i \in \set {1, 2, \ldots, n}$ such that:
 * $\lambda \in \map {\mathbb D} {a_{i i}, R_i}$

where:
 * $\displaystyle R_i = \sum_{j \mathop \ne i} \cmod {a_{ i j} }$
 * $\map {\mathbb D} {a, R}$ denotes the complex disk of center $a$ and radius $R$.