14 - Gerschgorin's 2nd Theorem

Gerschgorin's 2nd Theorem

Consider Gerschgorin Discs from Gerschgorin’s Theorem

If any union of $l$ discs is disjoint from the other discs then it contains exactly $l$ eigenvalues

Proof

Consider $B(\theta )=\theta A+(1-\theta )D$ where $D=\mathrm{diag}(A)$

As $\theta$ varies from $0$ to $1$ then $B(\theta )$ has entries varying from

$$B(0)=D\text{ to }B(1)=A$$

Hence eigenvalues vary continuously by Ostrowski’s Theorem

Gerschgorin Discs of $B(0)=D$ are points (diagonal entries) being eigenvalues of $D$
As $\theta$ increases then Gerschgorin Discs of $B(\theta )$ increase in radius about same points

Hence if $A=B(1)$ has a set of disjoint set of $l$ Gerschgorin Discs then
By continuity of eigenvalues it must contain exactly $l$ eigenvalues