 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2009/p/1/solution

A

in at least one disk

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

. To prove it, take a nonzero eigenvector

v

and choose

i

with

∣ v_{i} ∣ = max_{j} ∣ v_{j} ∣ > 0

. The

i

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

∣ v_{i} ∣

proves the disk bound.

Here

C = I

, and the three disks for

B + μ I

have real centres

2 + μ

3 + μ

4 + μ

2, 3, 4

respectively. Every point in every disk has real part at least

μ > 0

. Hence zero cannot be an eigenvalue, proving $B + μ C$ is nonsingular for every $μ > 0$ .

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