Explanation Procedure used: The expression a x 2 + b x y + c y 2 is the quadratic form associate with the quadratic equation a x 2 + b x y + c y 2 + d x + e y + f = 0 . A is the matrix of the quadratic form which is given by: A = [ a b 2 b 2 c ] Finding eigenvalues and eigenvectors: Let A be a n × n matrix. 1. Form the characteristics equation | λ I − A | = 0 . 2. Find the real roots of the characteristics equation. These are the eigenvalues of A . 3. For each eigenvalue λ i , find the eigenvectors corresponding to λ i by solving the homogeneous system ( λ i I − A ) X = 0 . This can be required row reducing a n × n matrix. Calculation: Consider the given equation: 5 x 2 − 2 x y + 5 y 2 + 10 x − 17 = 0 Compare with the general form of quadratic equation a x 2 + b x y + c y 2 + d x + e y + f = 0 , we get: a = 5 , b = − 2 and c = 5 . Substitute in the matrix of the quadratic form, we get: A = [ 5 − 2 2 − 2 2 5 ] = [ 5 − 1 − 1 5 ] The roots of the characteristic equation are: det ( [ 5 − 1 − 1 5 ] − λ [ 1 0 0 1 ] ) = 0 det ( [ 5 − 1 − 1 5 ] − [ λ 0 0 λ ] ) = 0 | 5 − λ − 1 − 1 5 − λ | = 0 ( 5 − λ ) ( 5 − λ ) − 1 = 0 Further simplify as: ( 5 − λ ) 2 − 1 = 0 ( 5 − λ ) = ± 1 λ = 4 , 6 Find the eigenvector corresponding to eigenvalue λ = 4 by solving the homogeneous system ( A − 4 I ) X = 0 : Let X = [ x 1 x 2 ] . ( [ 5 − 1 − 1 5 ] − 4 [ 1 0 0 1 ] ) [ x 1 x 2 ] = 0 ( [ 5 − 1 − 1 5 ] − [ 4 0 0 4 ] ) [ x 1 x 2 ] = 0 [ 1 − 1 − 1 1 ] [ x 1 x 2 ] = 0 It indicate as, x 1 − x 2 = 0

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN: 9781305658004

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 7.4, Problem 40E

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To find:

The matrix A of the quadratic form, eigenvalues and an orthogonal matrix P such that PTAP is diagonal from the equation 5x2−2xy+5y2+10x−17=0.

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Chapter 7.4, Problem 39E

Chapter 7.4, Problem 41E

Chapter 7 Solutions

Elementary Linear Algebra (MindTap Course List)

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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The matrix A of the quadratic form, eigenvalues and an orthogonal matrix P such that P T A P is diagonal from the equation 5 x 2 − 2 x y + 5 y 2 + 10 x − 17 = 0 .

Solution Summary: The author explains how to find the matrix A of the quadratic form, eigenvalues and an orthogonal matrix P.

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To find:

The matrix A of the quadratic form, eigenvalues and an orthogonal matrix P such that PTAP is diagonal from the equation 5x2−2xy+5y2+10x−17=0.

Want to see the full answer?

Chapter 7 Solutions

The matrix A of the quadratic form, eigenvalues and an orthogonal matrix P such that P T A P is diagonal from the equation 5 x 2 − 2 x y + 5 y 2 + 10 x − 17 = 0 .

Solution Summary: The author explains how to find the matrix A of the quadratic form, eigenvalues and an orthogonal matrix P.

Videos

To find: The matrix A of the quadratic form, eigenvalues and an orthogonal matrix P such that PTAP is diagonal from the equation 5x2−2xy+5y2+10x−17=0.

Want to see the full answer?

Chapter 7 Solutions

To find:

The matrix A of the quadratic form, eigenvalues and an orthogonal matrix P such that PTAP is diagonal from the equation 5x2−2xy+5y2+10x−17=0.