Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Related questions
Question
thumb_up100%
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](https://content.bartleby.com/qna-images/question/ebd9624c-6bf1-4607-b017-42b6f178ff82/4f15c88b-7c46-4467-90c5-9d4937e92094/a58zpde.png)
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
Expert Solution
arrow_forward
Step 1
Trending nowThis is a popular solution!
Step by stepSolved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Similar questions
- An environmental engineer wants to evaluate three different methods for disposing of nonhazardous chemical waste: land application, fluidized-bed incineration, and private disposal contract. Use the estimates below to help her determine which has the least cost at /= 15.00% per year on the basis of an annual worth evaluation. First Cost AOC per Year Salvage Value Life Land $-145000 $-86000 $23000 4 years Incineration $-760000 $-70000 $260000 6 years Contract O $-132000 O 2 yearsarrow_forward(c) Consider the following matrix. [3+31 Let B = A-¹. Find b₁2 (i.e., find the entry in row 1, column 2 of A¯¹) A = 1 - 3i 0 + 4i 3 + 3i 2 - 4iarrow_forwardGeneralize the result by proving that, for any comfortable non-singular matrices A,B and C the equation (ABC)-1 = C-1 B-1 A-1 holds.arrow_forward
- Consider matrix ?=(0−112)�=(01−12) and vectors ?=(1−2)�=(1−2) and ?=(11)�=(11). Calculate matrix ?=(?−??)2�=(�−��)2 and vector ?=??�=��. Then, evaluate the scalar product ?⋅?�⋅�.arrow_forward(c) Consider the following matrix. A = 3 - 3i 2 + 3i 2 + 4i 1 + 3i Let B = A-¹. Find b22 (i.e., find the entry in row 2, column 2 of A¯¹)arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
Numerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY 
Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,
Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,