hw7extra

hw7extra - f ( A ). This is called the spectral mapping...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
EE263 Prof. S. Boyd EE263 homework 7 additional exercise 1. Spectral mapping theorem. Suppose f : R R is analytic, i.e. , given by a power series expansion f ( u )= a 0 + a 1 u + a 2 u 2 + ··· (where a i = f ( i ) (0) / ( i !)). (You can assume that we only consider values of u for which this series converges.) For A R n × n ,wede±ne f ( A ) as f ( A )= a 0 I + a 1 A + a 2 A 2 + ··· (again, we’ll just assume that this converges). Suppose that Av = λv ,whe re v ± = 0, and λ C . Show that f ( A ) v = f ( λ ) v (ignoring the issue of convergence of series). We conclude that if λ is an eigenvalue of A ,then f ( λ ) is an
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: f ( A ). This is called the spectral mapping theorem . To illustrate this with an example, generate a random 3 3 matrix, for example using A=randn(3) . Find the eigenvalues of ( I + A )( I-A )-1 by rst computing this matrix, then nding its eigenvalues, and also by using the spectral mapping theorem. (You should get very close agreement; any dierence is due to numerical round-o errors in the various compua-tions.) 1...
View Full Document

Ask a homework question - tutors are online