Under what condition is the function harmonic? Here A = (aij ) is a given, symmetric 3 × 3 matrix.

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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
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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