26. Prove that A = 0 is an eigenvalue of A if and only if A is singular. 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 (x, x) = X(x, x). Hint: Recall that Ax = Ax. c. Show that ) = X; that is, the eigenvalue A is real. %3D
26. Prove that A = 0 is an eigenvalue of A if and only if A is singular. 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 (x, x) = X(x, x). Hint: Recall that Ax = Ax. c. Show that ) = X; that is, the eigenvalue A is real. %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question

Transcribed Image Text:26. Prove that A = 0 is an eigenvalue of A if and only if A is
singular.
||
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 X(x, x) = X(x, x). Hint: Recall that Ax = AX.
c. Show that l = X; that is, the eigenvalue A is real.
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