Gershgorin Circle Theorem

Author

Eric W. Weisstein
September 27, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/LinearAlgebra/GershgorinCircleTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GershgorinCircleTheorem.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

GerschgorinDisks[(a_List)?MatrixQ]:=Module[{abs=Abs[a],disks},disks=Transpose[{Through/@{Re,Im}/@Tr[a,List],(Plus@@#&)/@abs-Tr[abs,List]}];
{
{Hue[.5],Disk@@@disks},
Circle@@@disks
}
]

Graphic

Show[GraphicsArray[Partition[Graphics[{
GerschgorinDisks[#],
PointSize[.04],RGBColor[1,0,0],Point/@Through/@{Re,Im}/@Eigenvalues[#]
},AspectRatio->Automatic,Axes->True,Ticks->None]&/@Table[Random[Complex,{-20-20I,20+20I}],{8},{5},{5}],4]]]