Gershgorin Circle Theorem

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Theorem

Let $n$ be a positive integer.

Let $A = \left({a_{i j} }\right)$ be a complex square matrix of order $n$.

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


Then there exists $i \in \left\{ {1, 2, \ldots, n}\right\}$ such that:

$\lambda \in \mathbb D \left({a_{i i}, R_i}\right)$

where:

$\displaystyle R_i = \sum_{j \mathop \ne i} \left\vert{a_{ i j} }\right\vert$
$\mathbb D \left({a, R}\right)$ denotes the complex disk of center $a$ and radius $R$.


Proof


Source of Name

This entry was named for Semyon Aranovich Gershgorin.