**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 $[Q][Q]^T$.

The given quadratic form is given by F(x1,x2,x3 ) = 25 + 34 + 41 - 24

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 $[Q][Q]^T$.

Expert Solution

Step 1: Step 1

The given quadratic form is given by

F(x₁,x₂,x₃) = 25 $x subscript 1 superscript 2$ + 34 $x subscript 2 superscript 2$ + 41 $x subscript 3 superscript 2$ - 24 $x subscript 2 x subscript 3$

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