This means that krylov matrices tend to be ill

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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 Buy from 640 Chapter 7 Eigenvalues and Eigenvectors It is illegal to print, duplicate, or distribute this material Please report violations to 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...
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