Therefore the eigenvalues of a must be given by j b 2a

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Unformatted text preview: , and geo multB (1) = 1 = alg multB (1) , so B is diagonalizable. If An×n happens to have n distinct eigenvalues, then each eigenvalue is simple. This means that geo multA (λ) = alg multA (λ) = 1 for each λ, so (7.2.5) produces the following corollary guaranteeing diagonalizability. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 514 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Distinct Eigenvalues If no eigenvalue of A is repeated, then A is diagonalizable. Caution! The converse is not true—see Example 7.2.4. (7.2.6) It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Example 7.2.5 D E 72 Toeplitz matrices have constant entries on each diagonal parallel to the main diagonal. For example, a 4 × 4 Toeplitz matrix T along with a tridiagonal Toeplitz matrix A are shown below: ⎛ t0 ⎜ t−1 T=⎝ t−2 t−3 t1 t0 t−1 t−2 t2 t1 t0 t−1 ⎞ t3 t2 ⎟ ⎠, t1 t0 ⎛ T H t0 ⎜ t−1 A=⎝ 0 0 t1 t0 t−1 0 0 t1 t0 t−1 ⎞ 0 0⎟ ⎠. t1 t0 Toeplitz structures occur naturally in a variety of applications, and tridiagonal Toeplitz matrices are commonly the result of discretizing differential equation problems—e.g., see §1.4 (p. 18) and Example 7.6.1 (p. 559). The Toeplitz structure is rich in special properties, but tridiagonal Toeplitz matrices are particularly nice because they are among the few nontrivial structures that admit formulas for their eigenvalues and eigenvectors. IG R Y P Problem: Show that the eigenvalues and eigenvectors of ⎛ ⎞ ba ⎜c b ⎟ a ⎜ ⎟ .. .. .. ⎟ A=⎜ with a = 0 = c . . . ⎜ ⎟ ⎝ ⎠ c ba c b n×n O C are given by λj = b + 2a c/a cos jπ n+1 and ⎛ (c/a)1/2 sin (1jπ/(n + 1)) ⎞ ⎜ (c/a)2/2 sin (2jπ/(n + 1)) ⎟ ⎜ ⎟ 3 /2 xj = ⎜ (c/a) sin (3jπ/(n + 1)) ⎟ ⎜ ⎟ . ⎝ ⎠ . . (c/a)n/2 sin (njπ/(n + 1)) 72 Otto Toeplitz (1881–1940) was a professor in Bonn, Germany, but because of his Jewish background he was dismissed from his chair by the Nazis in 1933. In addition to the...
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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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