# Gershgorin Circle Theorem

Gershgorin Circle Theorem

This Demonstration illustrates how the Gershgorin circle theorem can be used to bound the eigenvalues of an matrix . The theorem states that the eigenvalues of must lie in circles defined in the complex plane that are centered on the diagonal elements of with radii determined by the row-norms of , that is, = and ={z∈:|z-|≤}. If of the circles form a connected region disjoint from the remaining circles, then the region contains exactly eigenvalues.

n×n

A

A

i

A

r

i

A

r

i

n

∑

j=1,j≠i

a

ij

i

a

ii

r

i

k

i

ℛ

n-k

ℛ

k

You can select a random matrix or a special matrix that has at least one disjoint Gershgorin circle. The elements of the random matrix, as well as some elements of the special matrix, can be changed by selecting a pseudorandom number generator from the dropdown menu. You can see either the separate Gershgorin circles with the eigenvalues or a plot that superimposes all the circles.

4×4

4×4