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4-3 3 Given A = (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 r¹ Ar, when x 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?

4-3 3 Given A = (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 r¹ Ar, when x 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
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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4-3 3 Given A = (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 € 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?
Transcribed Image Text:4-3 3 Given A = (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 € 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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