Explanation Approach: The procedure to find orthogonal diagonalization of a symmetric matrix : 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. Step-4: Check whether the matrix P is singular or nonsingular. Step -5: If the matrix P is nonsingular which implies that eigenvector is linear independent and A is diagonalizable. Step-6:If the matrix P is singular which implies that eigenvector is not linear independent and A is not diagonalizable. Step-7: The matrix P − 1 A P = P T A P = D will be diagonal.( The main diagonal entries of D are the eigenvalues of A ). Calculation: Consider the given matrix: A = [ 1 − 1 2 − 1 1 2 2 2 2 ] Apply the condition | A − λ I | = 0 for matrix A in order to calculate eigenvalues | [ 1 − 1 2 − 1 1 2 2 2 2 ] − λ [ 1 0 0 0 1 0 0 0 1 ] | = 0 | 1 − λ − 1 2 − 1 1 − λ 2 2 2 2 − λ | = 0 Expand the above determinant: ( 1 − λ ) ( λ 2 − 3 λ − 2 ) − ( − 1 ) ( λ − 6 ) + 2 ( 2 λ − 4 ) = 0 − λ 3 + 4 λ 2 + 4 λ − 16 = 0 − ( λ − 4 ) ( λ + 2 ) ( λ − 2 ) = 0 Equate the each product equal to zero: λ = − 2 , λ = 2 and λ = 4 Apply the | A − λ I | X = 0 in order to calculate eigenvector corresponding to eigenvalue λ = − 2 : [ 1 − ( − 2 ) − 1 2 − 1 1 − ( − 2 ) 2 2 2 2 − ( − 2 ) ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] [ 3 − 1 2 − 1 3 2 2 2 4 ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] Apply the transformation R 1 → 3 R 2 + R 1 in the above matrix: [ 3 − 1 2 − 1 3 2 2 2 4 ] ≡ [ 0 − 8 − 8 1 − 3 − 2 − 2 − 2 − 4 ] Apply the transformation R 3 → 2 R 2 + R 3 in the above matrix: [ 0 − 8 − 8 1 − 3 − 2 − 2 − 2 − 4 ] ≡ [ 0 − 8 − 8 1 − 3 − 2 0 − 8 − 8 ] Apply the transformation R 3 → − R 1 + R 3 and R 1 → − 1 8 R 1 in the above matrix: [ 0 − 8 − 8 1 − 3 − 2 0 − 8 − 8 ] ≡ [ 0 1 1 1 − 3 − 2 0 0 0 ] Apply the transformation R 2 → 3 R 1 + R 2 and R 1 ↔ R 2 in the above matrix: [ 0 1 1 1 − 3 − 2 0 0 0 ] ≡ [ 1 0 1 0 1 1 0 0 0 ] Write matrix notation of equation: [ 1 0 1 0 1 1 0 0 0 ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] Write above matrix notation in scalar form notation: x 1 + x 3 = 0 x 2 + x 3 = 0 Simplify further, x 1 = − x 3 x 2 = x 3 Let assume x 3 = s and substitute in the above equation, x 1 = − s x 2 = − s x 3 = s The eigenvector corresponding to eigenvalue λ = − 2 : v 1 = [ − s s s ] = s [ − 1 − 1 1 ] The eigenvector corresponding to eigenvalue λ = − 2 : v 1 = [ − 1 − 1 1 ] Apply the | A − λ I | X = 0 in order to calculate eigenvector corresponding to eigenvalue λ = 2 : [ 1 − ( 2 ) − 1 2 − 1 1 − ( 2 ) 2 2 2 2 − ( 2 ) ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] [ − 1 − 1 2 − 1 − 1 2 2 2 0 ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] Apply the transformation R 2 → R 2 − R 1 in the above matrix: [ − 1 − 1 2 − 1 − 1 2 2 2 0 ] ≡ [ 0 1 − 2 0 0 0 − 2 − 2 0 ] Apply the transformation R 3 → 2 R 2 + R 3 in the above matrix: [ 0 1 − 2 0 0 0 − 2 − 2 0 ] ≡ [ 1 1 − 2 0 0 0 0 0 − 4 ] Apply the transformation R 3 → − 1 4 R 3 and R 1 → R 1 + 2 R 3 in the above matrix: [ 1 1 − 2 0 0 0 0 0 − 4 ] ≡ [ 1 1 0 0 0 0 0 0 1 ] Apply the transformation R 3 ↔ R 2 in the above matrix: [ 1 1 0 0 0 0 0 0 1 ] ≡ [ 1 1 0 0 0 1 0 0 0 ] Write matrix notation of equation: [ 1 1 0 0 0 1 0 0 0 ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] Write above matrix notation in scalar form notation:

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 49E

To determine

To find:

A matrix P such that PTAP orthogonally diagonalizes A and verify that PTAP gives the correct diagonal form.

[1−12−112222]

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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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A matrix P such that P T A P orthogonally diagonalizes A and verify that P T A P gives the correct diagonal form. [ 1 − 1 2 − 1 1 2 2 2 2 ]

Solution Summary: The author explains the procedure to find orthogonal diagonalization of a symmetric matrix.

Videos

To find:

A matrix P such that PTAP orthogonally diagonalizes A and verify that PTAP gives the correct diagonal form.

[1−12−112222]

Chapter 7 Solutions

A matrix P such that P T A P orthogonally diagonalizes A and verify that P T A P gives the correct diagonal form. [ 1 − 1 2 − 1 1 2 2 2 2 ]

Solution Summary: The author explains the procedure to find orthogonal diagonalization of a symmetric matrix.

Videos

To find: A matrix P such that PTAP orthogonally diagonalizes A and verify that PTAP gives the correct diagonal form. [1−12−112222]

Chapter 7 Solutions

To find:

A matrix P such that PTAP orthogonally diagonalizes A and verify that PTAP gives the correct diagonal form.

[1−12−112222]