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


Taking the norm on both sides gives

Title Gershgorin’s circle theorem
Canonical name GershgorinsCircleTheorem
Date of creation 2013-03-22 13:14:15
Last modified on 2013-03-22 13:14:15
Owner lieven (1075)
Last modified by lieven (1075)
Numerical id 7
Author lieven (1075)
Entry type Theorem
Classification msc 15A42
Synonym Gershgorin’s disc theorem
Synonym Gerschgorin’s circle theorem
Synonym Gerschgorin’s disc theorem
Related topic BrauersOvalsTheorem
Defines Gershgorin disc
Defines Gerschgorin disc