# The desired similarity transformation is pnn j1 j2

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ∩ N (L) ⊆ R Li . A Jordan chain is built on top of each b ∈ Si by solving the system Li x = b for x and by setting IG R Jb = {Li x, Li−1 x, . . . , Lx, x}. Y P (7.7.2) Notice that chains built on top of vectors from Si each have length i + 1. The heuristic diagram in Figure 7.7.2 depicts Jordan chains built on top of the basis vectors illustrated in Figure 7.7.1—the chain that is labeled is built on top of a vector b ∈ S3 . O C Figure 7.7.2 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 577 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] The collection of vectors in all of these Jordan chains is a basis for C n . To demonstrate this, ﬁrst it must be argued that the total number of vectors in all Jordan chains is n, and then it must be proven that this collection is a linearly independent set. To count the number of vectors in all Jordan chains Jb , ﬁrst recall from (4.5.1) that the rank of a product is given by the formula rank (AB) = rank (B) − dim N (A) ∩ R (B), and apply this to conclude that dim Mi = dim R Li ∩ N (L) = rank Li − rank LLi . In other words, if we set di = dim Mi and ri = rank Li , then di = dim Mi = rank Li − rank Li+1 = ri − ri+1 , (7.7.3) D E so the number of vectors in Si is νi = di − di+1 = ri − 2ri+1 + ri+2 . (7.7.4) Since every chain emanating from a vector in Si contains i + 1 vectors, and since dk = 0 = rk , the total number of vectors in all Jordan chains is k−1 total = T H k−1 (i + 1)(di − di+1 ) (i + 1)νi = i=0 i=0 IG R = d0 − d1 + 2(d1 − d2 ) + 3(d2 − d3 ) + · · · + k (dk−1 − dk ) = d0 + d1 + · · · + dk−1 = (r0 − r1 ) + (r1 − r2 ) + (r2 − r3 ) + · · · + (rk−1 − rk ) = r0 = n. Y P To prove that the set of all vectors from all Jordan chains is linearly independent, place the...
View Full Document

## This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online