Under what condition is the function harmonic? Here A = (aij ) is a given, symmetric 3 × 3 matrix. ### Quadratic Form Representation in Matrix NotationConsider 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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Answered: Under what condition is the function…

Math

Advanced Math

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

Section: Chapter Questions

Problem 1RQ

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