hw7extra - f A This is called the spectral mapping theorem...

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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
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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 di²erence is due to numerical round-o² errors in the various compua-tions.) 1...
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This note was uploaded on 01/25/2012 for the course EE 263 taught by Professor Boyd,s during the Spring '08 term at Stanford.

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