Advanced Engineering Mathematics
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Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.](https://content.bartleby.com/qna-images/question/13b3ce4d-7939-40c7-a8c2-01af91160cea/1e85064e-6765-49e0-ac1d-89dcc87af720/81jbd4r_processed.png)
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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