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Unformatted text preview: of degree n with real or complex coefficients has n roots, but some roots may be complex numbers (even if all the coefficients are real), and some roots may be repeated. Consequently, A has n eigenvalues, but some may be complex, and some may be repeated. The fact that complex eigenvalues of real matrices must occur in conjugate pairs is a consequence of the fact that the roots of a polynomial with real coefficients occur in conjugate pairs. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.1 Elementary Properties of Eigensystems http://www.amazon.com/exec/obidos/ASIN/0898714540 493 Example 7.1.1 Problem: Determine the eigenvalues and eigenvectors of A = −1 1 1 1 . Solution: The characteristic polynomial is It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] det (A − λI) = 1−λ 1 −1 = (1 − λ)2 + 1 = λ2 − 2λ + 2, 1−λ so the characteristic equation is λ2 − 2λ + 2 = 0. Application of the quadratic formula yields √ √ 2 ± −4 2 ± 2 −1 λ= = = 1 ± i, 2 2 so the spectrum of A is σ (A) = {1 + i, 1 − i}. Notice that the eigenvalues are complex conjugates of each other—as they must be because complex eigenvalues of real matrices must occur in conjugate pairs. Now find the eigenspaces. For λ = 1 + i, D E A − λI = −i −1 1 −i −→ 1 0 i −1 1 i −→ 1 0 −i 0 IG R For λ = 1 − i, A − λI = T H =⇒ N (A − λI) = span i 0 =⇒ N (A − λI) = span i 1 . −i 1 . Y P In other words, the eigenvectors associated with λ1 = 1 + i are all nonzero T multiples of x1 = ( i 1 ) , and the eigenvectors associated with λ2 = 1 − i T are all nonzero multiples of x2 = ( −i 1 ) . In previous sections, you could be successful by thinking only in terms of real numbers and by dancing around those statements and issues involving complex numbers. But this example makes it clear that avoiding complex numbers, even when dealing with real matrices, is no longer possible—very innocent looking matrices, such as the one in this example, can...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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