Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / P / 4 / a

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- Solution a

Solution

The Gershgorin circle theorem states that every complex eigenvalue of a square matrix

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.

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