Given A = 5 (a) Find all the eigenvalues and eigenvectors of A. Are the eigenvalues real? Are the eigenvectors orthogonal? (b) Is A symmetric? If so, is A positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite? (c) What are the minimum and maximum values of the quadratic form z Ar, when a € R2 satisfies ||||= 1 but can otherwise be arbitrarily chosen? (d) Find the matrix norm ||A|| of A. What are the minimum and maximum gain input directions?
Given A = 5 (a) Find all the eigenvalues and eigenvectors of A. Are the eigenvalues real? Are the eigenvectors orthogonal? (b) Is A symmetric? If so, is A positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite? (c) What are the minimum and maximum values of the quadratic form z Ar, when a € R2 satisfies ||||= 1 but can otherwise be arbitrarily chosen? (d) Find the matrix norm ||A|| of A. What are the minimum and maximum gain input directions?
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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Question
parts C and D
Transcribed Image Text:Given A =
5
(a) Find all the eigenvalues and eigenvectors of A. Are the eigenvalues real? Are the eigenvectors
orthogonal?
(b) Is A symmetric? If so, is A positive definite, positive semidefinite, negative definite, negative
semidefinite, or indefinite?
(c) What are the minimum and maximum values of the quadratic form ¹ Ar, when re R² satisfies
||||= 1 but can otherwise be arbitrarily chosen?
(d) Find the matrix norm ||A|| of A. What are the minimum and maximum gain input directions?
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