# If x0 is a critical point then taylors theorem shows

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

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online