Eigenvalues of perturbed matrix. Rouché's theorem.

[SrEstroncio]

Homework Statement

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

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

Homework Equations

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

The Attempt at a Solution

I first expanding the \det 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.

 

[pasmith]

Homework Statement

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

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

Should that not be |z - \lambda_0| < \delta?

Homework Equations

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

The Attempt at a Solution

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