- #1

SrEstroncio

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## Homework Statement

Let [itex] \lambda_0 \in \mathbb{C} [/itex] be an eingenvalue of the [itex] n \times n [/itex] matrix [itex] A [/itex] with algebraic multiplicity [itex] m [/itex], that is, is an [itex]m[/itex]-nth zero of [itex] \det{A-\lambda I} [/itex]. Consider the perturbed matrix [itex] A+ \epsilon B [/itex], where [itex] |\epsilon | \ll 1 [/itex] and [itex] B [/itex] is any [itex] n \times n [/itex] matrix.

Show that given [itex] \delta \gt 0 [/itex], [itex] \alpha \gt 0 [/itex] exists so that, for [itex] | \epsilon | \lt \alpha [/itex], the matrix [itex] A + \epsilon B [/itex] has exactly [itex] m [/itex] eigenvalues (with algebraic multiplicity) inside [itex] | z - \lambda | \lt \delta [/itex]

## Homework Equations

Rouché's theorem states that if [itex] f [/itex] is holomorphic in a region and [itex] |g(z)| \lt |f(z)| [/itex] on a curve (suitable for integration) inside the open region, then [itex] f [/itex] and [itex] f+ g [/itex] have exactly the same amount of zeros (with multiplicity) inside the curve.

## The Attempt at a Solution

I first expanding the [itex] \det [/itex] function in power series and tried applying the integral which counts the number of zeros inside a region but I got stuck with several terms I couldn't get rid of. The TA told me to apply Rouché's theorem, but I can't figure out a way to exact the "sum" out of the determinant.

Any ideas? Any help would be appreciated.