Edu the manner in which we developed the jordan

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Unformatted text preview: pper-triangular matrix possessing a simple and predictable form. This turns out to be a fundamental problem, and the rest of this section is devoted to its solution. But before diving in, let’s set the stage by thinking about some possibilities. If P−1 LP = T is upper triangular, then the diagonal entries of T must be the eigenvalues of L, so T must have the form O C ⎛ ⎜ ⎜ T=⎜ ⎝ 0 .. . ··· .. . .. . ⎞ .⎟ .⎟ . ⎟. ⎠ 0 Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.7 Nilpotent Matrices and Jordan Structure http://www.amazon.com/exec/obidos/ASIN/0898714540 575 It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu One way to simplify the form of T is to allow nonzero entries only on the superdiagonal (the diagonal immediately above the main diagonal) of T, so we might try to construct a nonsingular P such that T has the form ⎛ ⎞ 0 .. .. ⎜ ⎟ . . ⎜ ⎟ T=⎜ ⎟. .. ⎝ ⎠ . 0 To gain some insight on how this might be accomplished, let L be a 3 × 3 nilpotent matrix for which L3 = 0 and L2 = 0, and search for a P such that ⎛ ⎞ ⎛ ⎞ 010 010 P−1 LP = ⎝ 0 0 1 ⎠ ⇐⇒ L[ P∗1 P∗2 P∗3 ] = [ P∗1 P∗2 P∗3 ] ⎝ 0 0 1 ⎠ 000 000 ⇐⇒ LP∗1 = 0, D E T H LP∗2 = P∗1 , LP∗3 = P∗2 . Since L3 = 0, we can set P∗1 = L2 x for any x3×1 such that L2 x = 0. This in turn allows us to set P∗2 = Lx and P∗3 = x. Because J = {L2 x, Lx, x} is a linearly independent set (Exercise 5.10.8), P = [ L2 x | Lx | x ] will do the job. J is called a Jordan chain, and it is characterized by the fact that its first vector is a somewhat special eigenvector for L while the other vectors are built (or “chained”) on top of this eigenvector to form a special basis for C 3 . There are a few more wrinkles in the development of a general theory for n × n nilpotent matrices, but the features illustrated here illuminate the path. For a general nilpotent matrix Ln×n = 0 of index k...
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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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