A problem from linear algebra: **Problem Statement:**14. Find a symmetric matrix $ Q $ such that $ q(\vec{x}) = \vec{x}^T Q \vec{x} = \vec{x}^T \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \vec{x} $.**Explanation:**The problem is to find a symmetric matrix $ Q $ such that the quadratic form $ q(\vec{x}) $ given by $ \vec{x}^T Q \vec{x} $ matches the quadratic form given by $ \vec{x}^T \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \vec{x} $.A matrix $ Q $ is symmetric if and only if $ Q = Q^T $. This means that the matrix should be equal to its transpose. \[\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\]The quadratic form $ q(\vec{x}) $ can be expanded as:\[\vec{x}^T Q \vec{x} = x_1 x_1 Q_{11} + x_1 x_2 (Q_{12} + Q_{21}) + x_1 x_3 (Q_{13} + Q_{31}) + x_2 x_2 Q_{22} + x_2 x_3 (Q_{23} + Q_{32}) + x_3 x_3 Q_{33}\]We need to equate this with:\[\vec{x}^T \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \vec{x}\]Which expands to:\[= 1 x_1^2 + (2 + 4)x_1 x_2 + (3 + 7)x_1 x_3 + 5 x_2^2 + (6 + 8)x_2 x_3 + 9 x_3^2\]So, \(

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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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A problem from linear algebra:

**Problem Statement:**

14. Find a symmetric matrix $ Q $ such that $ q(\vec{x}) = \vec{x}^T Q \vec{x} = \vec{x}^T \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \vec{x} $.

**Explanation:**

The problem is to find a symmetric matrix $ Q $ such that the quadratic form $ q(\vec{x}) $ given by $ \vec{x}^T Q \vec{x} $ matches the quadratic form given by $ \vec{x}^T \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \vec{x} $.

A matrix $ Q $ is symmetric if and only if $ Q = Q^T $. This means that the matrix should be equal to its transpose.

\[
\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}
\]

The quadratic form $ q(\vec{x}) $ can be expanded as:

\[
\vec{x}^T Q \vec{x} = x_1 x_1 Q_{11} + x_1 x_2 (Q_{12} + Q_{21}) + x_1 x_3 (Q_{13} + Q_{31}) + x_2 x_2 Q_{22} + x_2 x_3 (Q_{23} + Q_{32}) + x_3 x_3 Q_{33}
\]

We need to equate this with:

\[
\vec{x}^T \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \vec{x}
\]

Which expands to:

\[
= 1 x_1^2 + (2 + 4)x_1 x_2 + (3 + 7)x_1 x_3 + 5 x_2^2 + (6 + 8)x_2 x_3 + 9 x_3^2
\]

So, \(

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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