**Transforming Quadratic Forms: Finding the Change of Variable**To transform the quadratic form $ x^T A x $ into $ y^T D y $, we need to find the change of variable $ x = Py $. Below is the specific problem and the transformation involved:Given quadratic form:\[ -14x_1^2 - 13x_2^2 - 12x_3^2 + 24x_1x_2 - 24x_2x_3 \]We aim to transform it to:\[ 4y_1^2 - 13y_2^2 - 30y_3^2 \]The given expression demonstrates the initial quadratic form involving variables $ x_1, x_2, $ and $ x_3 $ and shows how it changes to involve new variables $ y_1, y_2, $ and $ y_3 $.### Explanation1. **Original Quadratic Form**: \[ -14x_1^2 - 13x_2^2 - 12x_3^2 + 24x_1x_2 - 24x_2x_3 \]2. **Transformed Quadratic Form**: \[ 4y_1^2 - 13y_2^2 - 30y_3^2 \]### Steps to Find the Change of Variable1. **Identify the Quadratic Form Matrix (A)**: - The original quadratic form can be represented as a symmetric matrix $ A $.2. **Diagonalize the Matrix (A)**: - Find the eigenvalues and eigenvectors of $ A $. 3. **Construct the Matrix (D)**: - (D) is the diagonal matrix consisting of the eigenvalues of $ A $.4. **Determine the Change of Basis Matrix (P)**: - (P) is composed of the eigenvectors of $ A $.5. **Transform Variables**: - Use the relation $ x = Py $ to transform the variables and obtain the new quadratic form.This transformation process helps in simplifying the quadratic form by converting it to a diagonal form, making it easier to analyze and solve problems related to quadratic expressions.

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Answered: Find the change of variable x = Py that…

Find the change of variable x = Py that transforms the quadratic form XTAX into yTDy as shown. - 14x3 - 13x3 - 12x3 + 24×,X2 - 24x,X3 = 4y? - 13y3 - 30y

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

**Transforming Quadratic Forms: Finding the Change of Variable**

To transform the quadratic form $ x^T A x $ into $ y^T D y $, we need to find the change of variable $ x = Py $. Below is the specific problem and the transformation involved:

Given quadratic form:
\[ -14x_1^2 - 13x_2^2 - 12x_3^2 + 24x_1x_2 - 24x_2x_3 \]

We aim to transform it to:
\[ 4y_1^2 - 13y_2^2 - 30y_3^2 \]

The given expression demonstrates the initial quadratic form involving variables $ x_1, x_2, $ and $ x_3 $ and shows how it changes to involve new variables $ y_1, y_2, $ and $ y_3 $.

### Explanation
1. **Original Quadratic Form**:
\[ -14x_1^2 - 13x_2^2 - 12x_3^2 + 24x_1x_2 - 24x_2x_3 \]

2. **Transformed Quadratic Form**:
\[ 4y_1^2 - 13y_2^2 - 30y_3^2 \]

### Steps to Find the Change of Variable
1. **Identify the Quadratic Form Matrix (A)**:
- The original quadratic form can be represented as a symmetric matrix $ A $.

2. **Diagonalize the Matrix (A)**:
- Find the eigenvalues and eigenvectors of $ A $.

3. **Construct the Matrix (D)**:
- (D) is the diagonal matrix consisting of the eigenvalues of $ A $.

4. **Determine the Change of Basis Matrix (P)**:
- (P) is composed of the eigenvectors of $ A $.

5. **Transform Variables**:
- Use the relation $ x = Py $ to transform the variables and obtain the new quadratic form.

This transformation process helps in simplifying the quadratic form by converting it to a diagonal form, making it easier to analyze and solve problems related to quadratic expressions.

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