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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
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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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