Consider the quadratic form F(X₁, X₂, X 3) = 25X² +34X² +41 X ² Write F - 24 X₂ X3 al F= [X² [A] [X WA b) Find a set of orthonormal eigenvectors for A where A is a symmetric matric 4) With the results of (b), form a normalized modal matri [Q] and determine the product [P][Q]
Consider the quadratic form F(X₁, X₂, X 3) = 25X² +34X² +41 X ² Write F - 24 X₂ X3 al F= [X² [A] [X WA b) Find a set of orthonormal eigenvectors for A where A is a symmetric matric 4) With the results of (b), form a normalized modal matri [Q] and determine the product [P][Q]
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
![**Quadratic Form Analysis**
Consider the quadratic form:
\[ F(x_1, x_2, x_3) = 25x_1^2 + 34x_2^2 + 41x_3^2 - 24x_2x_3 \]
a) Write \( F \) as:
\[ F = \begin{bmatrix} x \end{bmatrix}^T \begin{bmatrix} A \end{bmatrix} \begin{bmatrix} x \end{bmatrix} \]
where \( A \) is a symmetric matrix.
b) Find a set of orthonormal eigenvectors for \( A \).
c) With the results of (b), form a normalized modal matrix \([Q]\) and determine the product \([Q][Q]^T\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71f6a4b8-527f-4030-b85a-828002701c6f%2F07d0e458-692a-407b-b26b-6415f581dd9b%2Fgradas8_processed.png&w=3840&q=75)
Transcribed Image Text:**Quadratic Form Analysis**
Consider the quadratic form:
\[ F(x_1, x_2, x_3) = 25x_1^2 + 34x_2^2 + 41x_3^2 - 24x_2x_3 \]
a) Write \( F \) as:
\[ F = \begin{bmatrix} x \end{bmatrix}^T \begin{bmatrix} A \end{bmatrix} \begin{bmatrix} x \end{bmatrix} \]
where \( A \) is a symmetric matrix.
b) Find a set of orthonormal eigenvectors for \( A \).
c) With the results of (b), form a normalized modal matrix \([Q]\) and determine the product \([Q][Q]^T\).
Expert Solution
Step 1: Step 1
The given quadratic form is given by
F(x1,x2,x3 ) = 25
+ 34
+ 41
- 24
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