Page 89 - Matrices theory and applications

4. Norms
72
This result can also be deduced from Proposition 4.1.5: Let us decompose
A = D + C,where D is the diagonal part of A.If λ = a ii for every i,then
−1
C. Hence, if λ is an
λI n − A =(λI n − D)(I n − B)with B =(λI n − D)
eigenvalue, then either λ is an a ii ,or B ∞ ≥ 1.
One may improve this result by considering the connected components
of G.Let G be one of them. It is the union of the D k ’s that meet it. Let
p be the number of such disks. One then has G = ∪ i∈I D i where I has
cardinality p.
Theorem 4.5.1 There are exactly p eigenvalues of A in G,counted with
their multiplicities.
Proof
For r ∈ [0, 1], we define a matrix A(r) by the formula

a ii , j = i,
a ij (r):=
ra ij , j = i.
It is clear that the Gershgorin domain G r of A(r) is included in G.We
observe that A(1) = A,and that r → A(r) is continuous. Let us denote by
m(r) the number of eigenvalues (counted with multiplicity) of A(r)that
belong to G.
Since G and G\ G are compact, one can find a Jordan curve, oriented in
the trigonometric sense, that separates G from G\G. Let Γ be such a curve.
Since G r is included in G, the residue formula expresses m(r)in terms of
the characteristic polynomial of A(r):
1 P (z)

r
m(r)= dz.
2iπ P r (z)
Γ
Since P r does not vanish on Γ and r → P r ,P are continuous, we deduce

r
that r → m(r) is continuous. Since m(r) is an integer and [0, 1] is connected,
m(r) remains constant. In particular, m(0) = m(1).
Finally, m(0) is the number of entries a jj (eigenvalues of A(0)) that
belong to G.But a jj is in G if and only if D j ⊂ G. Hence m(0) = p,which
implies m(1) = p, the desired result.
An improvement of Gershgorin’s theorem concerns irreducible matrices.

Proposition 4.5.2 Let A be an irreducible matrix. If an eigenvalue of A
does not belong to the interior of any Gershgorin disk, then it belongs to

all the circles S(a ii ; |a ij |).
j =i
Proof
Let λ be such an eigenvalue and x an associated eigenvector. By assump-

tion, one has |λ − a ii |≥ |a ij | for every i.Let I be the set of indices
j =i