minimal Gershgorin set


related to Gershgorin’s theorem is the so called ”minimal Gershgorin set”. For every Aβˆˆπ‚n,n, 𝐱>0 meaning xi>0β€ƒβˆ€i, let’s define its minimal Gershgorin set G⁒(A) as:

G⁒(A)=⋂𝐱>0G𝐱⁒(A),

where

G𝐱⁒(A)=⋃i=1n{zβˆˆπ‚:|z-ai⁒i|≀1xiβ’βˆ‘jβ‰ i|ai⁒j|⁒xj}.

Theorem: Let Aβˆˆπ‚n,n, let σ⁒(A) be the spectrum of A and let G⁒(A) be its minimal Gershgorin set defined as above. Then

σ⁒(A)βŠ†G⁒(A).
Proof.

Given 𝐱>0, let X=d⁒i⁒a⁒g⁒{x1,x2,…,xn}and let BX=X-1⁒A⁒X. Then A and BX share the same spectrum, being similar. Due to definition, and keeping in mind that X-1=d⁒i⁒a⁒g⁒{x1-1,x2-1,…,xn-1}, we have bi⁒j(X)=ai⁒j⁒xjxi and, applying Gershgorin theorem to BX, we get:

σ⁒(A)=σ⁒(BX)βŠ†β‹ƒi=1n{zβˆˆπ‚:|z-ai⁒i|≀1xiβ’βˆ‘jβ‰ i|ai⁒j|⁒xj}

and, since this is true for any 𝐱>0, we finally get the thesis. ∎

Title minimal Gershgorin set
Canonical name MinimalGershgorinSet
Date of creation 2013-03-22 15:35:57
Last modified on 2013-03-22 15:35:57
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 11
Author Andrea Ambrosio (7332)
Entry type Definition
Classification msc 15A42