Gershgorin Circle Theorem

random matrix

special matrix

random generator method

MersenneTwister

random seed

1

Gershgorin circles and eigenvalues

superimposed circles

4 × 4 matrix and values

This Demonstration illustrates how the Gershgorin circle theorem can be used to bound the eigenvalues of an

n×n

matrix

A

. The theorem states that the eigenvalues of

A

must lie in circles



i

defined in the complex plane



that are centered on the diagonal elements of

A

with radii

r

i

determined by the row-norms of

A

, that is,

r

i

=

n

∑

j=1,j≠i



a

ij



and



i

={z∈:|z-

a

ii

|≤

r

i

}

. If

k

of the circles



i

form a connected region

ℛ

disjoint from the remaining

n-k

circles, then the region

ℛ

contains exactly

k

eigenvalues.

You can select a random

4×4

matrix or a special

4×4

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.

References

[1] B. Noble, Applied Linear Algebra, New Jersey: Prentice–Hall, 1969.

External Links

Gershgorin Circle Theorem (Wolfram MathWorld)

Gershgorin Circles

Brauer's Cassini Ovals versus Gershgorin Circles

Numerical Range for Some Complex Upper Triangular Matrices

Enclosing the Spectrum by Gershgorin-Type Sets

Permanent Citation

Housam Binous, Brian G. Higgins

"Gershgorin Circle Theorem"
http://demonstrations.wolfram.com/GershgorinCircleTheorem/
Wolfram Demonstrations Project
Published: April 6, 2012