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**Unformatted text preview: **hat is described in Example 7.9.4 (p. 606), and
describe f (A). Y
P 7.10.16. Limits and Group Inversion. Given a matrix Bn×n of rank r such
that index (B) ≤ 1 (i.e., index (λ = 0) ≤ 1 ), the Jordan form for B
0
looks like 0 C 0
= P−1 BP, so B = P 0 C P−1 , where C
0
0
r ×r
is nonsingular. This implies that B belongs to an algebraic group G
with respect to matrix multiplication, and the inverse of B in G is
B# = P 0 C0 1 P−1 . Naturally, B# is called the group inverse of
−
0
B. The group inverse is a special case of the Drazin inverse discussed in
Example 5.10.5 on p. 399, and properties of group inversion are developed in Exercises 5.10.11–5.10.13 on p. 402. Prove that if limk→∞ Ak
exists, and if B = I − A, then O
C lim Ak = I − BB# . k→∞ In other words, the limiting matrix can be characterized as the diﬀerence of two identity elements— I is the identity in the multiplicative
group of nonsingular matrices, and BB# is the identity element in the
multiplicative group containing B. Copyright c 2000 SIAM Buy online from SIAM
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7.10 Diﬀerence Equations, Limits, and Summability
http://www.amazon.com/exec/obidos/ASIN/0898714540 641 It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] 7.10.17. If Mn×n is a group matrix (i.e., if index (M) ≤ 1 ), then the group
inverse of M can be characterized as the unique solution M# of the
equations MM# M = M, M# MM# = M# , and MM# = M# M.
In fact, some authors use these equations to deﬁne M# . Use this characterization to show that if M = BC is any full-rank factorization of
∗
M, then M# = B(CB)−2 C. In particular, if M = U1 DV1 is the
full-rank factorization derived from the singular value decomposition as
described in (7.10.38), then
∗
∗
M# = U1 D−1/2 (V1 U1 )−2 D−1/2 V1 D
E ∗
∗
= U1 D−1 (V1 U1 )−2 V1
∗
∗
= U1 (V1 U1 )−2 D−1 V1 . T
H IG
R
Y
P O
C Copyright c 2000...

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