*This preview shows
page 1. Sign up to
view the full content.*

**Unformatted text preview: **for each > 0 there is a natural number
N = N ( ) such that vn − v < for all n ≥ N, and, by virtue of
Example 5.1.3 (p. 276), it doesn’t matter which norm is used. Therefore,
your proof should also be valid for vectors (and matrices). O
C 7.10.12. M-matrices Revisited. For matrices with nonpositive oﬀ-diagonal entries (Z-matrices), prove that the following statements are equivalent.
(a) A is an M-matrix.
(b) All leading principal minors of A are positive.
(c) A has an LU factorization, and both L and U are M-matrices.
(d) There exists a vector x > 0 such that Ax > 0.
(e) Each aii > 0 and AD is diagonally dominant for some diagonal matrix D with positive diagonal entries.
(f) Ax ≥ 0 implies x ≥ 0. Copyright c 2000 SIAM Buy online from SIAM
http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com
640
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540 It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu 7.10.13. Index by Full-Rank Factorization. Suppose that λ ∈ σ (A) , and
let M1 = A − λI. The following procedure yields the value of index (λ).
Factor M1 = B1 C1 as a full-rank factorization.
Set M2 = C1 B1 .
Factor M2 = B2 C2 as a full-rank factorization.
Set M3 = C2 B2 .
.
.
.
In general, Mi = Ci−1 Bi−1 , where Mi−1 = Bi−1 Ci−1 is a full-rank
factorization.
(a) Explain why this procedure must eventually produce a matrix
Mk that is either nonsingular or zero.
(b) Prove that if k is the smallest positive integer such that M−1
k
exists or Mk = 0, then
index (λ) = k−1
k D
E T
H if Mk is nonsingular,
if Mk = 0. 7.10.14. Use the procedure in Exercise 7.10.13 to ﬁnd the index of each eigenvalue
of A = −3
5
−1 −8
11
−2 −9
9
1 . Hint: σ (A) = {4, 1}. IG
R 7.10.15. Let A be the matrix given in Exercise 7.10.14.
(a) Find the Jordan form for A.
(b) For any function f deﬁned at A, ﬁnd the Hermite interpolation
polynomial t...

View Full
Document