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Unformatted text preview: tent decomposition as described on p. 397 (reordering the eigenvalues can put the nilpotent block Li on the bottom to realize the form in (5.10.5)). Consequently, the results in Example 5.10.3 (p. 398) insure that Pi Qi = Gi is the projector onto N (A − λi I)ki along R (A − λi I)ki , and this is true for all similarity transformations that reduce A to J. If A happens to be diagonalizable, then ki = 1 for each i, and the matrices Gi = Pi Qi are precisely the spectral projectors defined on p. 517. For this reason, there is no ambiguity in continuing to use the Gi notation, and we will continue to refer to the Gi ’s as spectral projectors. In the diagonalizable case, Gi projects onto the eigenspace associated with λi , and in the nondiagonalizable case Gi projects onto the generalized eigenspace associated with λi . Now consider ⎛ ⎞ D E ⎜ f (A) = Pf (J)P−1 = P ⎝ T H IG f J(λ1 ) .. ⎟ −1 ⎠P = R Y . f J(λs ) ⎛ Since f J(λi ) = ⎝ ⎞ .. . f J (λi ) .. . P s Pi f J(λi ) Qi . i=1 (7.9.5) ⎠ , where the J (λi ) ’s are the Jordan blocks associated with λi , (7.9.2) insures that if ki = index (λi ), then O C f J(λi ) = f (λi )I + f (λi )Li + f (λi ) 2 f (ki −1) (λi ) ki −1 , Li + · · · + L 2! (ki − 1)! i where Li = J(λi ) − λi I, and thus (7.9.5) becomes s f (A) = s ki −1 Pi f J(λi ) Qi = i=1 i=1 j =0 f (j ) (λi ) Pi Lj Qi . i j! (7.9.6) The terms Pi Lj Qi can be simplified by noticing that i ⎛Q ⎞ 1 P−1 P = I =⇒ Qi Pj = Copyright c 2000 SIAM I 0 if i = j, =⇒ P−1 Gi = if i = j, . ⎜.⎟ . Qi ⎠ Pi Qi ⎝. . . Qs ⎛0⎞ . . ⎜.⎟ = ⎝ Qi ⎠ , . . . 0 Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.9 Functions of Nondiagonalizable Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 and by using this with (7.9.4) to conclude that ⎛ j ⎜ ⎜ ⎜ (A − λi I) Gi = P ⎜ ⎜ ⎝ J(λ1 ) − λi I .. j . Lj i .. . J(λs ) − λi I 603 ⎞ ⎟ ⎟ ⎟ −1 j ...
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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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