**Unformatted text preview: **. . . , 0, where k = index (λ). T
H • Construct a basis B for N (A − λI).
Starting with any basis Sk−1 for Mk−1 (see p. 211), sequentially
extend Sk−1 with sets Sk−2 , Sk−3 , . . . , S0 such that
Sk−1
is a basis for Mk−1 ,
Sk−1 ∪ Sk−2
is a basis for Mk−2 ,
Sk−1 ∪ Sk−2 ∪ Sk−3 is a basis for Mk−3 , IG
R etc., until a basis B = Sk−1 ∪ Sk−2 ∪ · · · ∪ S0 = {b1 , b2 , . . . , bt }
for M0 = N (A − λI) is obtained (see Example 7.7.3 on p. 582).
• Y
P Build a Jordan chain on top of each eigenvector b ∈ B.
i
For each eigenvector b ∈ Si , solve (A − λI) x = b (a necessarily consistent system) for x , and construct a Jordan chain on
top of b by setting O
C i P = (A − λI) x i−1 (A − λI) x · · · (A − λI) x x (i+1)×n . Each such P corresponds to one Jordan block J (λ) in the Jordan segment J(λ) associated with λ.
The ﬁrst column in P is an eigenvector, and subsequent columns
are generalized eigenvectors of increasing order. • Copyright c 2000 SIAM If all such P ’s for a given λj ∈ σ (A) = {λ1 , λ2 , . . . , λs } are put in
a matrix Pj , and if P = P1 | P2 | · · · | Ps , then P is a nonsingular matrix such that P−1 AP = J = diag (J(λ1 ), J(λ2 ), . . . , J(λs ))
is in Jordan form as described on p. 590. Buy online from SIAM
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7.8 Jordan Form
http://www.amazon.com/exec/obidos/ASIN/0898714540 595 Example 7.8.3 Caution! Not every basis for N (A − λI) can be used to build Jordan chains
associated with an eigenvalue λ ∈ σ (A) . For example, the Jordan form of
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A = ⎝ −4
−4 0
1
0 ⎞
1
−2 ⎠
−1 ⎛ is 11
⎝0 1
J=
0 0 ⎞
0
0⎠
1 because σ (A) = {1} and index (1) = 2. Consequently, if P = [ x1 | x2 | x3 ]
is a nonsingular matrix such that P−1 AP = J, then the derivation beginning
on p. 593 leading to (7.8.5) shows that {x1 , x2...

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