Find the change of variable x = Py that transforms the quadratic form xTAx into yTDy as shown 2x? + 3x3 + 4x3 + 24x,×2 - 24×,x3 = 20y; + 3y? - 14y?
Find the change of variable x = Py that transforms the quadratic form xTAx into yTDy as shown 2x? + 3x3 + 4x3 + 24x,×2 - 24×,x3 = 20y; + 3y? - 14y?
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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![**Quadratic Form Transformation: Change of Variable**
To transform the quadratic form \( \mathbf{x}^T A \mathbf{x} \) into \( \mathbf{y}^T D \mathbf{y} \), find the change of variable \( \mathbf{x} = P \mathbf{y} \) as shown below:
\[ 2x_1^2 + 3x_2^2 + 4x_3^2 + 24x_1x_2 - 24x_2x_3 = 20y_1^2 + 3y_2^2 - 14y_2^3 \]
This equation represents the quadratic form transformation process, where \( \mathbf{x} \) and \( \mathbf{y} \) are vectors. The term \( \mathbf{x} \) is transformed by the matrix \( P \) to yield \( \mathbf{y} \), thereby converting the quadratic form \( \mathbf{x}^T A \mathbf{x} \) into \( \mathbf{y}^T D \mathbf{y} \). This transformation is essential for simplifying the quadratic form and making it easier to analyze. The given transformation details the coefficients of the quadratic terms and interaction terms both before and after the transformation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13b3ce4d-7939-40c7-a8c2-01af91160cea%2F1e85064e-6765-49e0-ac1d-89dcc87af720%2F81jbd4r_processed.png&w=3840&q=75)
Transcribed Image Text:**Quadratic Form Transformation: Change of Variable**
To transform the quadratic form \( \mathbf{x}^T A \mathbf{x} \) into \( \mathbf{y}^T D \mathbf{y} \), find the change of variable \( \mathbf{x} = P \mathbf{y} \) as shown below:
\[ 2x_1^2 + 3x_2^2 + 4x_3^2 + 24x_1x_2 - 24x_2x_3 = 20y_1^2 + 3y_2^2 - 14y_2^3 \]
This equation represents the quadratic form transformation process, where \( \mathbf{x} \) and \( \mathbf{y} \) are vectors. The term \( \mathbf{x} \) is transformed by the matrix \( P \) to yield \( \mathbf{y} \), thereby converting the quadratic form \( \mathbf{x}^T A \mathbf{x} \) into \( \mathbf{y}^T D \mathbf{y} \). This transformation is essential for simplifying the quadratic form and making it easier to analyze. The given transformation details the coefficients of the quadratic terms and interaction terms both before and after the transformation.
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