Under what condition is the function harmonic? Here A = (aij ) is a given, symmetric 3 × 3 matrix.
Under what condition is the function harmonic? Here A = (aij ) is a given, symmetric 3 × 3 matrix.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Under what condition is the function harmonic? Here A = (aij ) is a given, symmetric 3 × 3 matrix.
![### Quadratic Form Representation in Matrix Notation
Consider a quadratic function defined by:
\[ f(x_1, x_2, x_3) = \sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} x_i x_j \]
Here, \(A = (a_{ij})\) is a given, symmetric \(3 \times 3\) matrix.
### Explanation:
1. \( f(x_1, x_2, x_3) \): This represents a quadratic function in three variables \(x_1\), \(x_2\), and \(x_3\).
2. \(\sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} x_i x_j\): These are double summations indicating that for each combination of \(i\) and \(j\) from 1 to 3, the term \(a_{ij} x_i x_j\) is summed.
3. \(a_{ij}\): Each \((i, j)\) element of the matrix \(A\).
4. \(A = (a_{ij})\): A symmetric matrix, meaning that \(a_{ij} = a_{ji}\). This symmetry is crucial for many properties and calculations involving quadratic forms.
### Matrix Explanation:
A symmetric \(3 \times 3\) matrix \(A\) can be represented as:
\[
A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{12} & a_{22} & a_{23} \\
a_{13} & a_{23} & a_{33}
\end{pmatrix}
\]
In this matrix:
- \(a_{11}\), \(a_{22}\), and \(a_{33}\) are the diagonal elements.
- \(a_{12} = a_{21}\), \(a_{13} = a_{31}\), \(a_{23} = a_{32}\) demonstrate the symmetry of the matrix.
### Visualization:
To understand how this quadratic form translates into operation on vectors, one might interpret it as follows:
Given a vector \(\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\) and the matrix \(A\), the quadratic](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febd9624c-6bf1-4607-b017-42b6f178ff82%2F4f15c88b-7c46-4467-90c5-9d4937e92094%2Fa58zpde.png&w=3840&q=75)
Transcribed Image Text:### Quadratic Form Representation in Matrix Notation
Consider a quadratic function defined by:
\[ f(x_1, x_2, x_3) = \sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} x_i x_j \]
Here, \(A = (a_{ij})\) is a given, symmetric \(3 \times 3\) matrix.
### Explanation:
1. \( f(x_1, x_2, x_3) \): This represents a quadratic function in three variables \(x_1\), \(x_2\), and \(x_3\).
2. \(\sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} x_i x_j\): These are double summations indicating that for each combination of \(i\) and \(j\) from 1 to 3, the term \(a_{ij} x_i x_j\) is summed.
3. \(a_{ij}\): Each \((i, j)\) element of the matrix \(A\).
4. \(A = (a_{ij})\): A symmetric matrix, meaning that \(a_{ij} = a_{ji}\). This symmetry is crucial for many properties and calculations involving quadratic forms.
### Matrix Explanation:
A symmetric \(3 \times 3\) matrix \(A\) can be represented as:
\[
A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{12} & a_{22} & a_{23} \\
a_{13} & a_{23} & a_{33}
\end{pmatrix}
\]
In this matrix:
- \(a_{11}\), \(a_{22}\), and \(a_{33}\) are the diagonal elements.
- \(a_{12} = a_{21}\), \(a_{13} = a_{31}\), \(a_{23} = a_{32}\) demonstrate the symmetry of the matrix.
### Visualization:
To understand how this quadratic form translates into operation on vectors, one might interpret it as follows:
Given a vector \(\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\) and the matrix \(A\), the quadratic
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