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3. Let A be an n x n matrix. Let (x, y) = x"y be the scalar product for R". (a) Show that (Ax, y) = (x, A" y) for all x, y € R". (b) Suppose that A is symmetric, i.e., AT = A. Suppose that x and y are eigenvectors satisfying Ax = \x and Ay = µy with A, µ e R and A # µ. Show that (x, y) = 0, i.e., show that x I y.

3. Let A be an n x n matrix. Let (x, y) = x"y be the scalar product for R". (a) Show that (Ax, y) = (x, A" y) for all x, y € R". (b) Suppose that A is symmetric, i.e., AT = A. Suppose that x and y are eigenvectors satisfying Ax = \x and Ay = µy with A, µ e R and A # µ. Show that (x, y) = 0, i.e., show that x I y.

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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3. Let A be an n x n matrix. Let (x, y) = x'y be the scalar product for R". (a) Show that (Ax, y) = (x, ATy) for all x, y € R". (b) Suppose that A is symmetric, i.e., A" = A. Suppose that x and y are eigenvectors satisfying Ax show that x ly. Ax and Ay = µy with A, µ E R and A # µ. Show that (x, y) = 0, i.e.,
Transcribed Image Text:3. Let A be an n x n matrix. Let (x, y) = x'y be the scalar product for R". (a) Show that (Ax, y) = (x, ATy) for all x, y € R". (b) Suppose that A is symmetric, i.e., A" = A. Suppose that x and y are eigenvectors satisfying Ax show that x ly. Ax and Ay = µy with A, µ E R and A # µ. Show that (x, y) = 0, i.e.,
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