Eigenvalues of perturbed matrix. Rouché's theorem.

  • #1

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.
 

Answers and Replies

  • #2
pasmith
Homework Helper
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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]

Should that not be [itex]|z - \lambda_0| < \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.

I think you want to take
[tex]f(z) = \det (A - zI)[/tex]
and
[tex]g(z,\epsilon) = \det (A + \epsilon B - zI) - f(z)[/tex]
and then look at the curve [itex]|z - \lambda_0| = \delta[/itex]. If you can show that, for all [itex]\delta > 0[/itex], there exists [itex]\alpha > 0[/itex] such that for all [itex]|\epsilon| < \alpha[/itex], [itex]|g(z,\epsilon)| < |f(z)|[/itex] on that curve, then Rouché's theorem will give you the result.
 

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