Back to the index. Or to the chambers

This article has 13 links. View as Cloud or List.

Proof Chu Space Diagonal Summation Normed Vector Space Complex Basic Polygon Frame Matrix Radius Proof Of Uniqueness Of Center Of A Circle Thin Square Eigenvalue Of A Linear Operator

Loading ...

Planetmath Browser (2008—2009)

BSD licence | A django site

All articles taken from PlanetMath.org snapshot under CC-BY-SA licence.

→ The original article on PlanetMath.org

Other Formats: LaTeX

New! You can click on formulas to copy the LaTeX source to your clipboard. | Math Videos

Gershgorins Circle Theorem

Let be a square complex matrix. Around every element $a_{ii}$ 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 $\sum_{j\neq i}\vert a_{ij}\vert$ . Such circles are called Gershgorin discs.

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

Proof: Let $\lambda$ be an eigenvalue of and its corresponding eigenvector. Choose such that $\vert x_i\vert={\max}_j \vert x_j\vert$ . Since can't be 0, $\vert x_i\vert>0$ . Now $Ax=\lambda x$ , or looking at the -th component

$\displaystyle (\lambda - a_{ii})x_i = \sum_{j\neq i} a_{ij}x_j.$

Taking the norm on both sides gives

$\displaystyle \vert\lambda - a_{ii}\vert=\vert\sum_{j\neq i} \frac{a_{ij}x_j}{x_i}\vert\leq \sum_{j\neq i}\vert a_{ij}\vert.$

1	Diagonal
1	Basic Polygon
1	Matrix
1	Frame
1	Complex
1	Summation
1	Proof
1	Chu Space
1	Normed Vector Space
2	Radius
2	Proof Of Uniqueness Of Center Of A Circle
3	Thin Square
5	Eigenvalue Of A Linear Operator