Page 88 - Matrices theory and applications

Since this holds true for every p, we conclude that
lim sup v m ≤ inf v p ≤ lim inf v p ,
which proves the convergence to the infimum.
Characterization (complex case). If R< 1/r(x), the Taylor series

m m
z ∈ CC,
z x , 4.5. The Gershgorin Domain 71
m∈IN
converges in norm in the ball B(0; R). Its sum equals (e − zx) −1 (see
the proof of Proposition 4.1.5).
The domain of the map z → (e − zx) −1 is open, since if it contains a
point z 0 , the previous paragraph shows that e − (z − z 0 )(e − z 0 x) −1 x
is invertible for every z satisfying
−1
|z − z 0 |r (e − z 0x) x < 1.
Denoting by X z the inverse, we see that X z (e − z 0 x) −1 is an inverse
of e − zx.In particular, f : z → (e − z) −1 is holomorphic.
If f is defined on a ball B(0; s), Cauchy’s formula
1 (m) 1 f(z)
m
x = f (0) = dz
m! 2iπ B(0;s) z m+1
m
shows that x = O(s −m ). Hence, 1/r(x) ≥ s.
Corollary 4.4.1 Let B ∈ M n (K) be given. Then B m m→+∞ 0 if and only
→
if ρ(B) < 1.
m
m
Indeed, ρ(B) ≥ 1 implies B ≥ ρ(B ) ≥ 1 for every m.Conversely,
m
ρ(B) < 1 implies B <r m for m large enough, where r is selected in
(ρ(B), 1).
We observe that this result is also a consequence of Householder’s
theorem.

4.5 The Gershgorin Domain

Let A ∈ M n (CC), and let λ be an eigenvalue and x an associated eigenvector.
Let i be an index such that |x i | = x ∞ .Then x i =0 and

x j
|a ii − λ| = a ij ≤ |a ij |.

j =i x i j =i

Proposition 4.5.1 (Gershgorin) The spectrum of A is included in the
Gershgorin domain G(A), defined as the union of the Gershgorin disks
D i := D(a ii ; |a ij |).
j =i