**Unformatted text preview: **ectors
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7.7.9. Prove that if N is the Jordan form for a nilpotent matrix L as described
in (7.7.5) and (7.7.6) on p. 579, then for any set of nonzero scalars
˜
{ 1 , 2 , . . . , t } , the matrix L is similar to a matrix N of the form
⎛ 1 N1 It is illegal to print, duplicate, or distribute this material
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˜
N=⎜ .
⎝.
.
0 0 ···
2 N2 · · ·
..
.
0 ··· ⎞
0
0⎟
. ⎟.
.⎠
.
t Nt D
E In other words, the 1’s on the superdiagonal of the Ni ’s in (7.7.5) are
artiﬁcial because any nonzero value can be forced onto the superdiagonal
of any Ni . What’s important in the “Jordan structure” of L is the
number and sizes of the nilpotent Jordan blocks (or chains) and not the
values appearing on the superdiagonals of these blocks. T
H IG
R Y
P O
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7.8 Jordan Form
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7.8 JORDAN FORM 587 The goal of this section is to do for general matrices A ∈ C n×n what was done for
nilpotent matrices in §7.7—reduce A by means of a similarity transformation
to a block-diagonal matrix in which each block has a simple triangular form.
The two major components for doing this are now in place—they are the corenilpotent decomposition (p. 397) and the Jordan form for nilpotent matrices. All
that remains is to connect these two ideas. To do so, it is convenient to adopt
the following terminology. D
E Index of an Eigenvalue The index of an eigenvalue λ for a matrix A ∈ C n×n is deﬁned to
be the index of the matrix (A − λI) . In other words, from the characterizations of index given on p. 395, index (λ) is the smallest positive
integer k such that any one of the following statements is true.
•
• R (A − λI)k = R (A − ...

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