If x0 is a critical point then taylors theorem shows

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Unformatted text preview: ote that L is the two-dimensional version of the one-dimensional finite-difference matrix in Example 1.4.1 (p. 19). D E Problem: Show L is positive definite by explicitly exhibiting its eigenvalues. T H Solution: Example 7.2.5 (p. 514) insures that the n eigenvalues of T are λi = 4 − 2 cos iπ n+1 , i = 1, 2, . . . , n. IG If U is an orthogonal matrix such that UT TU = D = diag and if B is the n2 × n2 block-diagonal orthogonal matrix ⎛ D −I ⎛ ⎞ U 0 ··· 0 ⎜ −I D ⎜ 0 U ··· 0 ⎟ ⎜ .. ˜ B=⎜ . . .. . ⎟, then BT LB = L = ⎜ . ⎜ ⎝. . . .⎠ . . . ⎝ 0 0 ··· U R Y P (7.6.7) (λ1 , λ2 , . . . , λn ) , ⎞ −I .. . −I .. . D −I ⎟ ⎟ ⎟. ⎟ −I ⎠ D Consider the permutation obtained by placing the numbers 1, 2, . . . , n2 rowwise in a square matrix, and then reordering them by listing the entries columnwise. For example, when n = 3 this permutation is generated as follows: ⎛ ⎞ 123 ˜ v = (1, 2, 3, 4, 5, 6, 7, 8, 9) → A = ⎝ 4 5 6 ⎠ → (1, 4, 7, 2, 5, 8, 3, 6, 9) = v. 789 O C Equivalently, this can be described in terms of wrapping and unwrapping rows by wrap unwrap writing v−−→A −→ AT − − v. If P is the associated n2 × n2 permutation − − −→ ˜ matrix, then ⎞ ⎛ λi −1 ⎞ ⎛ T1 0 · · · 0 ⎟ ⎜ −1 λi −1 ⎟ ⎜ ⎜ 0 T2 · · · 0 ⎟ .. .. .. T˜ ⎟. ⎟ with Ti = ⎜ ⎜. P LP = ⎝ . . .⎠ .. . . . ⎟ ⎜ . . . . . . ⎝ −1 λi −1 ⎠ 0 0 · · · Tn −1 λi n×n Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 566 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 If you try it on the 9 × 9 case, you will see why it works. Now, Ti is another tridiagonal Toeplitz matrix, so Example 7.2.5 (p. 514) again applies to yield σ (Ti ) = {λi − 2 cos (jπ/n + 1) , j = 1, 2, . . . , n} . This together with (7.6.7) produces the n2 eigenvalues of L as It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] λij = 4 − 2 cos iπ n+1 + cos jπ n+1 , i, j...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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