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27. In this problem we show that the eigenvalues of a Hermitian matrix A are real. Let x be an eigenvector corresponding to the eigenvalue A. a. Show that (Ax, x) = (x, Ax). Hint: See Problem 21c. b. Show that A(x, x) = X(x, x). Hint: Recall that Ax = Ax. c. Show that A = X; that is, the eigenvalue A is real. 28. Show that if A1 and A2 are eigenvalues of a Hermitian matrix A, and if A1 # A2, then the corresponding eigenvectors x() and x(2) are orthogonal. Hint: Use the results of Problems 21c and 27 to show that (A1 – 12)(x(), x(2)) = 0. -

27. In this problem we show that the eigenvalues of a Hermitian matrix A are real. Let x be an eigenvector corresponding to the eigenvalue A. a. Show that (Ax, x) = (x, Ax). Hint: See Problem 21c. b. Show that A(x, x) = X(x, x). Hint: Recall that Ax = Ax. c. Show that A = X; that is, the eigenvalue A is real. 28. Show that if A1 and A2 are eigenvalues of a Hermitian matrix A, and if A1 # A2, then the corresponding eigenvectors x() and x(2) are orthogonal. Hint: Use the results of Problems 21c and 27 to show that (A1 – 12)(x(), x(2)) = 0. -

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Hint: 21. a. Suppose that A is a real-valued n x n matrix. Show that (Ax, y) = (x, A' y) for any vectors x and y. Hint: You may find it simpler to consider first the case n = 2; then extend the result to an arbitrary value of n. b. If A is not necessarily real, show that (Ax, y) = (x, A*y) for any vectors x and y. c. If A is Hermitian, show that (Ax, y) vectors x and y. (x, Ay) for any
Transcribed Image Text:Hint: 21. a. Suppose that A is a real-valued n x n matrix. Show that (Ax, y) = (x, A' y) for any vectors x and y. Hint: You may find it simpler to consider first the case n = 2; then extend the result to an arbitrary value of n. b. If A is not necessarily real, show that (Ax, y) = (x, A*y) for any vectors x and y. c. If A is Hermitian, show that (Ax, y) vectors x and y. (x, Ay) for any
27. In this problem we show that the eigenvalues of a Hermitian matrix A are real. Let x be an eigenvector corresponding to the eigenvalue A. a. Show that (Ax, x) = (x, Ax). Hint: See Problem 21c. b. Show that A(x, x) = X(x, x). Hint: Recall that Ax = Ax. c. Show that A = X; that is, the eigenvalue A is real. 28. Show that if A1 and A2 are eigenvalues of a Hermitian matrix A, and if A1 # A2, then the corresponding eigenvectors x() and x(2) are orthogonal. Hint: Use the results of Problems 21c and 27 to show that (A1 – 12)(x(), x(2)) = 0. -
Transcribed Image Text:27. In this problem we show that the eigenvalues of a Hermitian matrix A are real. Let x be an eigenvector corresponding to the eigenvalue A. a. Show that (Ax, x) = (x, Ax). Hint: See Problem 21c. b. Show that A(x, x) = X(x, x). Hint: Recall that Ax = Ax. c. Show that A = X; that is, the eigenvalue A is real. 28. Show that if A1 and A2 are eigenvalues of a Hermitian matrix A, and if A1 # A2, then the corresponding eigenvectors x() and x(2) are orthogonal. Hint: Use the results of Problems 21c and 27 to show that (A1 – 12)(x(), x(2)) = 0. -
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