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Unformatted text preview: i = j without A being normal.
7.5.8. Explain why a triangular matrix is normal if and only if it is diagonal. It is illegal to print, duplicate, or distribute this material
Please report violations to firstname.lastname@example.org 7.5.9. Use the result of Exercise 7.5.8 to give an alternate proof of the unitary
diagonalization theorem given on p. 547. D
E 7.5.10. For a normal matrix A, explain why (λ, x) is an eigenpair for A if
and only if (λ, x) is an eigenpair for A∗ . 7.5.11. To see if you understand the proof of the min-max part of the Courant–
Fischer theorem (p. 550), construct an analogous proof for the max-min
part of (7.5.5). T
H 7.5.12. The Courant–Fischer theorem has the following alternate formulation.
λi = max v1 ,...,vn−i ∈C n min x∗ Ax IG
and x⊥v1 ,...,vn−i
x 2 =1 λi = min v1 ,...,vi−1 ∈C n max x∗ Ax x⊥v1 ,...,vi−1
x 2 =1 for 1 < i < n. To see if you really understand the proof of the minmax part of (7.5.5), adapt it to prove the alternate min-max formulation
7.5.13. (a) Y
P Explain why every unitary matrix
onal matrix of the form
⎛ i θ1
⎜ 0 e i θ2
C Copyright c 2000 SIAM (b) 0 0 is unitarily similar to a diag···
. · · · e i θn Prove that every orthogonal matrix is orthogonally similar to a
real block-diagonal matrix of the form
cos θ1 sin θ1
− sin θ1 cos θ1
cos θt sin θt
− sin θt cos θt Buy online from SIAM
http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com
Eigenvalues and Eigenvectors
7.6 POSITIVE DEFINITE MATRICES It is illegal to print, duplicate, or distribute this material
Please report violations to email@example.com Since the symmetric structure of a matrix forces its eigenvalues to be real, what
additional property will force all eigenvalues to be p...
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