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axp xp1 xp o c 2 3 p in other words the rst column

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Unformatted text preview: ectors http://www.amazon.com/exec/obidos/ASIN/0898714540 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 Please report violations to [email protected] ⎜0 ˜ 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 C Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Buy from AMAZON.com 7.8 Jordan Form http://www.amazon.com/exec/obidos/ASIN/0898714540 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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