In fact since a is a tridiagonal toeplitz matrix the

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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 meyer@ncsu.edu 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 R 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 given above. 7.5.13. (a) Y P Explain why every unitary matrix onal matrix of the form ⎛ i θ1 e 0 ⎜ 0 e i θ2 D=⎜ . . ⎝. . . . O C Copyright c 2000 SIAM (b) 0 0 is unitarily similar to a diag··· ··· .. . ⎞ 0 0⎟ . ⎟. .⎠ . · · · e i θn Prove that every orthogonal matrix is orthogonally similar to a real block-diagonal matrix of the form ⎛ ⎞ ±1 .. ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ±1 ⎜ ⎟ ⎜ ⎟ cos θ1 sin θ1 ⎟. B=⎜ ⎜ ⎟ − 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 558 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 7.6 POSITIVE DEFINITE MATRICES It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu 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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