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**Unformatted text preview: **· Bk ).
(A ⊗ B)∗ = A∗ + B∗ .
rank (A ⊗ B) = (rank (A))(rank (B)).
Assume A is m × m and B is n × n for the following.
trace (A ⊗ B) = (trace (A))(trace (B)).
(A ⊗ In )(Im ⊗ B) = A ⊗ B = (Im ⊗ B)(A ⊗ In ).
det (A ⊗ B) = (det (A))n (det (B))m .
(A ⊗ B)−1 = A−1 ⊗ B−1 .
Let the eigenvalues of Am×m be denoted by λi and let the eigenvalues
of Bn×n be denoted by µj . Prove the following. D
E T
H IG
R n
◦ The eigenvalues of A ⊗ B are the mn numbers {λi µj }im j =1 .
=1 Y
P n
◦ The eigenvalues of (A ⊗ In ) + (Im ⊗ B) are {λi + µj }im j =1 .
=1 7.8.12. Use part (b) of Exercise 7.8.11 along with the result of Exercise 7.6.10
(p. 573) to construct an alternate derivation of (7.6.8) on p. 566. That
is, show that the n2 eigenvalues of the discrete Laplacian Ln2 ×n2 described in Example 7.6.2 (p. 563) are given by O
C λij = 4 sin2 iπ
2(n + 1) + sin2 jπ
2(n + 1) , i, j = 1, 2, . . . , n. Hint: Recall Exercise 7.2.18 (p. 522). 7.8.13. Determine the eigenvalues of the three-dimensional discrete Laplacian
by using the formula from Exercise 7.6.10 (p. 573) that states
Ln3 ×n3 = (In ⊗ In ⊗ An ) + (In ⊗ An ⊗ In ) + (An ⊗ In ⊗ In ). Copyright c 2000 SIAM Buy online from SIAM
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7.9 Functions of Nondiagonalizable Matrices
http://www.amazon.com/exec/obidos/ASIN/0898714540
7.9 FUNCTIONS OF NONDIAGONALIZABLE MATRICES 599 It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] The development of functions of nondiagonalizable matrices parallels the development for functions of diagonal matrices that was presented in §7.3 except that
the Jordan form is used in place of the diagonal matrix of eigenvalues. Recall
from the discussion surrounding (7.3.5) on p. 526 that if A ∈ C n×n is diagonalizable, say A = PDP−1 , where D = diag (λ1 I, λ2 I, . . . , λs I) , and if f (λi )
exists for each λi , then f (A) is deﬁned to...

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