Explanation Approach: The procedure to find the eigenvalue and eigenvector are, Step-1: Apply the condition | A − λ I | = 0 for matrix A in order to calculate eigenvalues: Step-2: Apply the | A − λ I | X = 0 in order to calculate eigenvectors u 1 and u 2 corresponding to eigenvalue λ 1 and λ 2 respectively. Step-3: Construct the matrix P whose column are the eigenvector. If there are two matrix A = [ a i j ] r 1 × c 1 and B = [ b i j ] r 2 × c 2 A = [ a 11 a 12 a 13 a 21 a 22 a 23 ] 2 × 3 and B = [ b 11 b 12 b 21 b 22 b 31 b 33 ] 3 × 2 The multiplication of two given matrix: A B = [ a 11 a 12 a 13 a 21 a 22 a 23 ] 2 × 3 [ b 11 b 12 b 21 b 22 b 31 b 33 ] 3 × 2 = [ a 11 ( b 11 ) a 12 ( b 21 ) a 13 ( b 31 ) a 21 ( b 12 ) a 22 ( b 22 ) a 23 ( b 33 ) ] 3 × 3 A square matrix A with a real element said to be orthogonal if it satisfied following condition 1: A T = A − 1 2: A A T = A T A = I The set of column vector { P 1 , P 2 , P 3 , … P n } said to be orthogonal if it satisfied following condition. P i ⋅ P j = 0 , i ≠ j Calculation: Consider the given symmetry matrix: A = [ 1 0 1 0 1 0 0 0 − 1 ] Apply the condition | A − λ I | = 0 for matrix A in order to calculate eigenvalues | [ 1 0 1 0 1 0 0 0 − 1 ] − λ [ 1 0 0 0 1 0 0 0 1 ] | = 0 | 1 − λ 0 1 0 1 − λ 0 0 0 − 1 − λ | = 0 Expand the above determinant: ( 1 − λ ) ( 1 − λ ) ( 1 + λ ) = 0 ( 1 − λ ) 2 ( 1 + λ ) = 0 Equate the each product equal to zero: λ = 1 , λ = − 1 and λ = 1 Apply the | A − λ I | X = 0 in order to calculate eigenvector corresponding to eigenvalue λ = 1 : [ 1 − 1 0 1 0 1 − 1 0 0 0 − 1 − 1 ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] [ 0 0 1 0 0 0 0 0 − 2 ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] Write above matrix notation in scalar form notation: 0 x 1 + 0 x 2 + 1 x 3 = 0 0 x 1 + 0 x 2 + 0 x 3 = 0 0 x 1 + 0 x 2 − 2 x 3 = 0 Let assume x 1 = s and x 2 = t

8th Edition

ISBN: 9780357156100

Author: Ron Larson

Publisher: Cengage Limited

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Chapter 7.3, Problem 38E

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The whether eigenvector of the given symmetric matrix corresponding to distinct eigenvalue are orthogonal.

[10101000−1]

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Chapter 7 Solutions

Elementary Linear Algebra

Ch. 7.1 - Verifying Eigenvalues and Eigenvectors in...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Verifying Eigenvalues and Eigenvectors in...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Determining Eigenvectors In Exercise 9-12,...Ch. 7.1 - Determining Eigenvectors In Exercise 9-12,...

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To check: The whether eigenvector of the given symmetric matrix corresponding to distinct eigenvalue are orthogonal. [ 1 0 1 0 1 0 0 0 − 1 ]

Solution Summary: The author explains that the given symmetric matrix corresponding to distinct eigenvalue are orthogonal.

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To check:

The whether eigenvector of the given symmetric matrix corresponding to distinct eigenvalue are orthogonal.

[10101000−1]

Chapter 7 Solutions

To check: The whether eigenvector of the given symmetric matrix corresponding to distinct eigenvalue are orthogonal. [ 1 0 1 0 1 0 0 0 − 1 ]

Solution Summary: The author explains that the given symmetric matrix corresponding to distinct eigenvalue are orthogonal.

Videos

To check: The whether eigenvector of the given symmetric matrix corresponding to distinct eigenvalue are orthogonal. [10101000−1]

Chapter 7 Solutions

To check:

The whether eigenvector of the given symmetric matrix corresponding to distinct eigenvalue are orthogonal.

[10101000−1]