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?

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section: Chapter Questions
Problem 9RQ
icon
Related questions
Question

parts a and b

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?
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?
Expert Solution
steps

Step by step

Solved in 4 steps with 4 images

Blurred answer
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,