Theorem of Gershgorin

In numerical Analysis, the theorem of Gershgorin is a result making it possible to limit the eigenvalues a priori of a square matrix. It was published in 1931 by the Belorusse Mathématicien Semion Aranovitch Gershgorin.

The theorem

Statement

Either has a complex matrix of size N × N , of general term ( has _ij). For each index of line I between 1, and N one introduces the disc of Gershgorin corresponding

D_i= \ left \ {Z \ in \ mathbb {C}, |a_ {II} - Z|\ Leq \ sum_ {J \ neq I}|a_ {ij}| \ right \} =D (a_ {II}, R_i)

who constitutes indeed a Disque in the complex plan, of ray R_i .

Theorem : any eigenvalue of has belongs to at least of the discs of Gershgorin.

By applying the theorem to the Matrice transposed of has , new information is given on the localization of the eigenvalues: they are in the meeting of the discs of Gershgorin associated with the columns

\ tilde {D} _j= \ left \ {Z \ in \ mathbb {C}, |a_ {jj} - Z|\ Leq \ sum_ {I \ neq J}|a_ {ij}| \ right \} =D (a_ {jj}, \ tilde {R} _j)

Demonstration

Are λ a Eigenvalue of has and X an associated clean vector, noted components ( X _J). They check the relations

(\ lambda - a_ {II}) x_i = \ sum_ {J \ neq I} a_ {ij} x_j

for I ranging between 1 and N . By choosing an index I for which the module of X _I is maximum. Since X is vector clean, | X _I| is nonnull and it is possible to form the quotient

|a_ {II} - \ lambda| = \ left|\ sum_ {J \ neq I} a_ {ij} \ frac {x_j} {x_i} \ right|\ Leq \ sum_ {J \ neq I} |a_ {ij}|

See too

References

Gerschgorin, S. " Über die Abgrenzung der Eigenwerte einer Matrix." Izv. Akad. Nauk. US Otd. Fiz. - Chechmate. Nauk 7,749-754, 1931
Varga, R.S. Geršgorin and His Circles. Berlin: Springer-Verlag, 2004. ISBN 3-540-21100-4. Errata.

External bonds

Eric W. Weisstein. " Gershgorin Circle Theorem." in the MathWorld site--With Wolfram Web Resource.

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