Make a change of variable, x = Py, that transforms the quadratic form 3x, + 14x, x, + 3x, into a quadratic form with no cross-product term. Give P and the new quadratic form. Which of the following matrices can be used as P in x = Py to produce a quadratic form with no cross product term? OB. Oc. VD. 1 1 V2 V2 1 1 1 1 1 V2 12 The corresponding quadratic form is y' Dy = ||| (Simplify your answer.) |- |-
Make a change of variable, x = Py, that transforms the quadratic form 3x, + 14x, x, + 3x, into a quadratic form with no cross-product term. Give P and the new quadratic form. Which of the following matrices can be used as P in x = Py to produce a quadratic form with no cross product term? OB. Oc. VD. 1 1 V2 V2 1 1 1 1 1 V2 12 The corresponding quadratic form is y' Dy = ||| (Simplify your answer.) |- |-
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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Question
![**Problem Statement:**
Make a change of variable, \( x = P y \), that transforms the quadratic form \( 3x_1^2 + 14x_1x_2 + 3x_2^2 \) into a quadratic form with no cross-product term. Give \( P \) and the new quadratic form.
**Choose the Correct Matrix:**
Which of the following matrices can be used as \( P \) in \( x = P y \) to produce a quadratic form with no cross-product term?
**Options:**
- **Option A:**
\[
\begin{bmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
\end{bmatrix}
\]
- **Option B:**
\[
\begin{bmatrix}
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{bmatrix}
\]
- **Option C:**
\[
\begin{bmatrix}
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{bmatrix}
\]
- **Option D (Correct Answer):**
\[
\begin{bmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
\end{bmatrix}
\]
**Solution:**
The corresponding quadratic form is \( y^T D y = \boxed{11(y_1^2 + y_2^2)} \). (Simplify your answer.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e6b254a-d25d-479c-b42b-752b8b5bbcbb%2F77eb4de7-4b58-45a8-9cf0-c074983f4679%2F6nv6goh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Make a change of variable, \( x = P y \), that transforms the quadratic form \( 3x_1^2 + 14x_1x_2 + 3x_2^2 \) into a quadratic form with no cross-product term. Give \( P \) and the new quadratic form.
**Choose the Correct Matrix:**
Which of the following matrices can be used as \( P \) in \( x = P y \) to produce a quadratic form with no cross-product term?
**Options:**
- **Option A:**
\[
\begin{bmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
\end{bmatrix}
\]
- **Option B:**
\[
\begin{bmatrix}
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{bmatrix}
\]
- **Option C:**
\[
\begin{bmatrix}
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{bmatrix}
\]
- **Option D (Correct Answer):**
\[
\begin{bmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
\end{bmatrix}
\]
**Solution:**
The corresponding quadratic form is \( y^T D y = \boxed{11(y_1^2 + y_2^2)} \). (Simplify your answer.)
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