Explanation Given: The matrix is, A = [ 2 3 0 − 1 0 1 − 2 − 1 4 ] Approach: For a given n × n matrix A , the polynomial p ( λ ) defined by p ( λ ) = det ( A − λ I ) is called the characteristic polynomial of A , and the equation p ( λ ) = 0 is called the characteristic equation of A . The polynomial of the form, p ( λ ) = ( − 1 ) n ( λ − λ 1 ) m 1 ( λ − λ 2 ) m 2 ⋯ ( λ − λ k ) m k Thus, associated with each eigenvalue λ i is a number m i , called the multiplicity of λ i . Let A be an n × n matrix. For a given eigenvalue λ i , let E i denotes the set of all vectors v satisfying A v = λ i v . Then E i is called the eigenspace of a A corresponding to the eigenvalue λ i . Thus, E i is the solution set to the linear system ( A − λ i I ) v = 0 . If n i denotes the dimension of E i , then 1 ≤ n i ≤ m i for each i . In words, the dimension of the eigenspace corresponding to λ i is at most the multiplicity of λ i . An n × n matrix A that has n linearly independent eigenvectors is called nondefective. In such a case, we say that A has a complete set of eigenvectors. If A has less than n linearly independent eigenvectors, it is called defective. Calculation: Calculate the eigenvalues for the matrix A . The characteristic polynomial of A can be written as, p ( λ ) = | 2 − λ 3 0 − 1 − λ 1 − 2 − 1 4 − λ | = ( 2 − λ ) [ − λ ( 4 − λ ) + 1 ] − 3 [ − 1 ( 4 − λ ) + 2 ] = ( λ − 2 ) 3 The characteristic equation of A is, ( λ − 2 ) 3 = 0 λ = 2 The eigenvalues for the matrix A are λ 1 = 2 . The polynomial of the eigenvalues can be written as, p ( λ ) = ( λ − 2 ) 3 Since the polynomial have power 3 then, Thus, the multiplicity of the eigenvalues λ 1 is 3 . Calculate the eigenvector corresponding to eigenvalue λ 1 = 2 . ( A − λ i I ) v = 0 The augmented matrix for the system is, [ 0 3 0 0 − 1 − 2 1 0 − 2 − 1 2 0 ] Applying the elementary row operation, R 1 ↔ R 3

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Question

Chapter 7.2, Problem 15P

To determine

The multiplicity of each eigenvalue, basis for each eigenspace, dimension of each eigenspace of the given matrix A and whether the matrix is defective or nondefective.

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Chapter 7.2, Problem 14P

Chapter 7.2, Problem 16P

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Chapter 7 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 7.1 - Prob. 1P Ch. 7.1 - Prob. 2P Ch. 7.1 - Prob. 3P Ch. 7.1 - Prob. 4P Ch. 7.1 - Prob. 5P Ch. 7.1 - Given that v1=(2,1) and v2=(1,1) are eigenvectors...Ch. 7.1 - Prob. 7P Ch. 7.1 - Prob. 8P Ch. 7.1 - Prob. 9P Ch. 7.1 - Prob. 11P

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