Gershgorin circle theorem

In mathematics, the Gershgorin circle theorem (also called Gershgorin Disk Theorem) may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. Gershgorin's name has been transliterated in several different ways, including Geršgorin, Gerschgorin, Gershgorin, Hershhorn, and Hirschhorn.

Statement and proof

Let

A

{\displaystyle A}

be a complex

n

×

n

{\displaystyle n\times n}

matrix, with entries

a

i

j

{\displaystyle a_{ij}}

. For

i

∈

{

1

,

…

,

n

}

{\displaystyle i\in \{1,\dots ,n\}}

, let

R

i

{\displaystyle R_{i}}

be the sum of the absolute values of the non-diagonal entries in the

i

{\displaystyle i}

-th row:

R

i

=

∑

j

≠

i

|

a

i

j

|

.

{\displaystyle R_{i}=\sum _{j\neq {i}}\left|a_{ij}\right|.}

Let

D

(

a

i

i

,

R

i

)

⊆

C

{\displaystyle D(a_{ii},R_{i})\subseteq \mathbb {C} }

be a closed disc centered at

a

i

i

{\displaystyle a_{ii}}

with radius

R

i

{\displaystyle R_{i}}

. Such a disc is called a Gershgorin disc.

Theorem. Every eigenvalue of

A

{\displaystyle A}

lies within at least one of the Gershgorin discs

D

(

a

i

i

,

R

i

)

.

{\displaystyle D(a_{ii},R_{i}).}

Proof.