That is ig r y p 0 0 jtjj j1 0

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Unformatted text preview: lues of A. For example, if the Jordan form for A is ⎛ O C ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ J=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 4 ⎞ 10 41 4 4 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 1 4 3 0 1 3 2 2 then we know that A9×9 has three distinct eigenvalues, namely σ (A) = {4, 3, 2}; alg mult (4) = 5, alg mult (3) = 2, and alg mult (2) = 2; geo mult (4) = 2, geo mult (3) = 1, and geo mult (2) = 2; Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.8 Jordan Form http://www.amazon.com/exec/obidos/ASIN/0898714540 593 index (4) = 3, index (3) = 2, and index (2) = 1; λ = 2 is a semisimple eigenvalue, so, while A is not diagonalizable, part of it is; i.e., the restriction A/ is a diagonalizable linear operator. N (A−2I) It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Of course, if both P and J are known, then A can be completely reconstructed from (7.8.4), but the point being made here is that only J is needed to reveal the eigenstructure along with the other similarity invariants of A. Now that the structure of the Jordan form J is known, the structure of the similarity transformation P such that P−1 AP = J is easily revealed. Focus on a single p × p Jordan block J (λ) contained in the Jordan segment J(λ) associated with an eigenvalue λ, and let P = [ x1 x2 · · · xp ] be the portion of P = [ · · · | P | · · ·] that corresponds to the position of J (λ) in J. Notice that AP = PJ implies AP = P J (λ) or, equivalently, ⎛ ⎜ ⎜ A[ x1 x2 · · · xp ] = [ x1 x2 · · · xp ] ⎜ ⎝ λ D E T H 1 .. . ⎞ . .. IG R .. . ⎟ ⎟ ⎟, 1⎠ λ p×p so equating columns on both sides of this equation produces Ax1 = λx1 =⇒ x1 is an eigenvector =⇒ (A − λI) x1 = 0, Ax2 = x1 + λx2 =⇒ (A − λI) x2 = x1 =⇒ (A − λI) x2 = 0, =⇒ (A − λI) x3 = x2 . . . =⇒ (A − λI) x3 = 0, . . . =⇒ (A − λI) xp = xp−1 =⇒ (A − λI) xp = 0. Y P Ax3 = x2...
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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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