Explanation Given: The given matrix is: [ 0 − 1 1 0 ] Procedure used: Finding the eigenvalues and eigenvectors: Expanding the determinant of | A − r I | = 0 and then, solve the n th–degree polynomial in r . Therefore, finding the eigenvalues is equal to the finding the zero of the polynomial. The above expression is called characteristics equation. The corresponding nontrivial solutions r are called eigenvectors and It get by solving the ( A − r I ) u = 0 Here, u = [ u 1 u 2 ] . Calculation: Consider the matrix: A = [ 0 − 1 1 0 ] Substitute the value of matrix A in the characteristic equation and solve for r : | [ 0 − 1 1 0 ] − r [ 1 0 0 1 ] | = 0 | [ 0 − 1 1 0 ] − [ r 0 0 r ] | = 0 | [ − r − 1 1 − r ] | = 0 r 2 + 1 = 0 Take first factor: r 2 + 1 = 0 r = ± i Therefore, the eigenvalues are r 1 = i and r 2 = − i . Find the eigenvector corresponding to eigenvalue r 1 = i by using the formula ( A − r I ) u = 0 : ( A − r I ) u = ( [ 0 − 1 1 0 ] − i [ 1 0 0 1 ] ) [ u 1 u 2 ] = ( [ 0 − 1 1 0 ] − [ i 0 0 i ] ) [ u 1 u 2 ] = [ − i − 1 1 − i ] [ u 1 u 2 ] The matrix equation is: [ − i − 1 1 − i ] [ u 1 u 2 ] = [ 0 0 ] It can be write in scalar equation as, − i u 1 − u 2 = 0

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 9.5, Problem 9E

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The eigenvalues and eigenvectors for the given matrix.

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Chapter 9.5, Problem 8E

Chapter 9.5, Problem 10E

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Fundamentals of Differential Equations and Boundary Value Problems

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The eigenvalues and eigenvectors for the given matrix.

Solution Summary: The author explains how to find the eigenvalues and enumerates the nontrivial solutions for the given matrix.

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To find: The eigenvalues and eigenvectors for the given matrix.

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Chapter 9 Solutions

To find:

The eigenvalues and eigenvectors for the given matrix.