778 in which nj l v j jb 0 0 j

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Unformatted text preview: ⎞ ⎛ z0 ⎜ z2 ⎟ = [ 0 | 0 | · · · | Lk−1 Qk−1 ] ⎜ . ⎟ =⇒ zk−1 ∈ N Lk−1 Qk−1 ⎝.⎠ . O C Y P =⇒ zk−1 zk−1 = 0. This conclusion with the same argument applied to 0 = Lk−2 Qz produces zk−2 = 0. Similar repetitions show that zi = 0 for each i, and thus N (Q) = 0. It has now been proven that if B = Sk−1 ∪ Sk−2 ∪ · · · ∪ S0 = {b1 , b2 , . . . , bt } is the basis for N (L) derived from the nested subspaces Mi , then the set of all Jordan chains J = Jb1 ∪ Jb2 ∪ · · · ∪ Jbt is a basis for C n . If the vectors from J are placed as columns (in the order in which they appear in J ) in a matrix Pn×n = [ J1 | J2 | · · · | Jt ], then P is nonsingular, and if bj ∈ Si , then Jj = [ Li x | Li−1 x | · · · | Lx | x ] for some x such that Li x = bj so that ⎛ ⎞ 01 .. .. ⎜ ⎟ . . ⎜ ⎟ LJj = [ 0 | Li x | · · · | Lx ] = [ Li x | · · · | Lx | x ] ⎜ . . 1 ⎟ = Jj Nj , ⎝ ⎠ . 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 579 where Nj is an (i + 1) × (i + 1) matrix whose entries are equal to superdiagonal and zero elsewhere. Therefore, ⎛ N1 0 · · · ⎜ 0 N2 · · · LP = [ LJ1 | LJ2 | · · · | LJt ] = [ J1 | J2 | · · · | Jt ] ⎜ . .. ⎝. . . 0 1 along the 0 0 . . . ⎞ ⎟ ⎟ ⎠ · · · Nt 0 It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu or, equivalently, ⎛ N1 0 · · · ⎜ 0 N2 · · · P−1 LP = N = ⎜ . .. ⎝. . . 0 0 ··· ⎛ ⎞ 0 0 ⎜ 0⎟ ⎜ . ⎟ , where Nj = ⎜ .⎠ ⎝ . Nt 1 .. . ⎞ .. . .. . D E ⎟ ⎟ ⎟ . (7.7.5) 1⎠ 0 T H Each Nj is a nilpotent matrix whose index is given by its size. The Nj ’s are called nilpotent Jordan blocks, and the block-diagonal matrix N is called the Jordan form for L. Below is a summary. IG R Jordan Form for a Nilpotent Matrix Every nilpotent matrix Ln×n of index k is similar to a bloc...
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