 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2007/p/4/a/solution

A

lies in at least one closed disc

D_{i} = {z : ∣ z - a_{ii} ∣ \leq \sum_{j \neq = i} ∣ a_{ij} ∣}

. To prove it, take a nonzero eigenvector

v

for

λ

and choose

i

with

∣ v_{i} ∣

maximal. The

i

th row of

A v = λ v

gives

∣ λ - a_{ii} ∣ ∣ v_{i} ∣ = \sum_{j \neq = i} a_{ij} v_{j} \leq \sum_{j \neq = i} ∣ a_{ij} ∣ ∣ v_{j} ∣ \leq (\sum_{j \neq = i} ∣ a_{ij} ∣) ∣ v_{i} ∣.

∣ v_{i} ∣ > 0

proves the claim. The proof applies to real and complex matrices alike.

 New to topics? Read the docs here!