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 columns 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 = [ 3 3 3 3 ] Apply the condition | A − λ I | = 0 for matrix A in order to calculate eigenvalues | [ 3 3 3 3 ] − λ [ 1 0 0 1 ] | = 0 | 3 − λ 3 3 3 − λ | = 0 Expand the above determinant: ( 3 − λ ) ( 3 − λ ) − 9 = 0 ( 3 − λ ) 2 − 3 2 = 0 ( 3 − λ − 3 ) ( 3 − λ + 3 ) = 0 λ ( 6 − λ ) = 0 Equate the each product equal to zero: λ = 0 and λ = 6 Apply the | A − λ I | X = 0 in order to calculate eigenvector corresponding to eigenvalue λ = 0 :

Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + WebAssign Printed Access Card for Larson's Elementary Linear Algebra, 8th Edition, Single-Term

8th Edition

ISBN: 9781337604925

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 7.3, Problem 33E

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Whether eigenvector of the symmetric matrix [3333] corresponding to distinct eigenvalues are orthogonal.

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Chapter 7.3, Problem 32E

Chapter 7.3, Problem 34E

Chapter 7 Solutions

Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + WebAssign Printed Access Card for Larson's Elementary Linear Algebra, 8th Edition, Single-Term

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: Whether eigenvector of the symmetric matrix [ 3 3 3 3 ] corresponding to distinct eigenvalues are orthogonal.

Solution Summary: The author explains the procedure to calculate eigenvalues and enumerates the columns of the matrix P.

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To check:

Whether eigenvector of the symmetric matrix [3333] corresponding to distinct eigenvalues are orthogonal.

Want to see the full answer?

Chapter 7 Solutions

To check: Whether eigenvector of the symmetric matrix [ 3 3 3 3 ] corresponding to distinct eigenvalues are orthogonal.

Solution Summary: The author explains the procedure to calculate eigenvalues and enumerates the columns of the matrix P.

Videos

To check: Whether eigenvector of the symmetric matrix [3333] corresponding to distinct eigenvalues are orthogonal.

Want to see the full answer?

Chapter 7 Solutions

To check:

Whether eigenvector of the symmetric matrix [3333] corresponding to distinct eigenvalues are orthogonal.