Gershgorin Circle Theorem

From ProofWiki
Jump to navigation Jump to search

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 $R$.


Proof


Source of Name

This entry was named for Semyon Aranovich Gershgorin.


Sources