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单词 GershgorinsCircleTheorem
释义

Gershgorin’s circle theorem


Let A be a square complex matrix. Around every element aii on the diagonal of the matrix, we draw a circle with radius the sum of the norms of the other elements on the same row ji|aij|. Such circles are called Gershgorin discs.

Theorem: Every eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A lies in one of these Gershgorin discs.

Proof: Let λ be an eigenvalue of A and x its corresponding eigenvectorMathworldPlanetmathPlanetmathPlanetmath. Choose i such that |xi|=maxj|xj|. Since x can’t be 0, |xi|>0. Now Ax=λx, or looking at the i-th componentPlanetmathPlanetmathPlanetmath

(λ-aii)xi=jiaijxj.

Taking the norm on both sides gives

|λ-aii|=|jiaijxjxi|ji|aij|.
TitleGershgorin’s circle theorem
Canonical nameGershgorinsCircleTheorem
Date of creation2013-03-22 13:14:15
Last modified on2013-03-22 13:14:15
Ownerlieven (1075)
Last modified bylieven (1075)
Numerical id7
Authorlieven (1075)
Entry typeTheorem
Classificationmsc 15A42
SynonymGershgorin’s disc theorem
SynonymGerschgorin’s circle theorem
SynonymGerschgorin’s disc theorem
Related topicBrauersOvalsTheorem
DefinesGershgorin disc
DefinesGerschgorin disc
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