Past exam of the options II course of the University of Oxford / 2007 / P / P4 / a

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- Solution a

Solution

Gershgorin's theorem states that every complex eigenvalue of a square matrix lies in at least one row disc

D_{i} = {z \in C : ∣ z - a_{ii} ∣ \leq R_{i}}, R_{i} = \sum_{j \neq = i} ∣ a_{ij} ∣.

Let

A v = λ v

with

v \neq = 0

, and choose an index

i

where

∣ v_{i} ∣

is maximal. Its row equation gives

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

Dividing by the positive

∣ v_{i} ∣

proves

λ \in D_{i}

. This proof applies to complex eigenvectors even when the matrix is real.

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