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:

$\ds R_i = \sum_{j \mathop \ne i} \cmod {a_{ i j} }$

$\map {\mathbb D} {a, R}$ denotes the complex disk of center $a$ and radius $R$.

Proof

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Source of Name

This entry was named for Semyon Aranovich Gershgorin.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Gerschgoren circle theorem
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Gerschgorin's Theorem