Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
1. Consider the matrices -3 0 2-2 A₁ ¹ = ( ₁ ) ₁4₂ = ( −8 −2 ) ₁4₁ = ( −² − ²). 4 1 13 , , A3 0 -2 -2 2 These matrices can be used to build functions x¹ A₁x, x¹ A₂x, x¹ A3x, and x¹ A4r. These functions are represented (in an arbitrary order) in Figure 1. Match each matrix with its corresponding picture. (a) -5 -4 and 4₁ = (-). A4 -4 Explain briefly. (b) (c) Figure 1: Graphical representation of functions €

1. Consider the matrices -3 0 2-2 A₁ ¹ = ( ₁ ) ₁4₂ = ( −8 −2 ) ₁4₁ = ( −² − ²). 4 1 13 , , A3 0 -2 -2 2 These matrices can be used to build functions x¹ A₁x, x¹ A₂x, x¹ A3x, and x¹ A4r. These functions are represented (in an arbitrary order) in Figure 1. Match each matrix with its corresponding picture. (a) -5 -4 and 4₁ = (-). A4 -4 Explain briefly. (b) (c) Figure 1: Graphical representation of functions €

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please help

# Matrix-Function Relationships ## Consider the Matrices \[ A_1 = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix}, \quad A_2 = \begin{pmatrix} -3 & 0 \\ 0 & -2 \end{pmatrix}, \quad A_3 = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}, \quad \text{and} \quad A_4 = \begin{pmatrix} -5 & -4 \\ -4 & 0 \end{pmatrix}. \] These matrices can be used to build functions \( x^T A_1 x, x^T A_2 x, x^T A_3 x, \) and \( x^T A_4 x \). These functions are represented (in an arbitrary order) in Figure 1. Match each matrix with its corresponding picture. ### Figure 1: Graphical Representation of Functions - **(a)** A surface plot showing a hyperbolic paraboloid shape spanning a positive and negative space, often associated with matrices having both positive and negative eigenvalues. - **(b)** A surface plot depicting a relatively flat plane that descends uniformly, typically representative of negative definite matrices. - **(c)** A surface plot with a dome-like upward curvature, usually indicating a positive definite matrix. - **(d)** A surface plot showing a saddle shape, indicative of an indefinite matrix with both positive and negative eigenvalues. In summary, these graphs illustrate the functions derived from the quadratic forms \( x^T A x \) for the given matrices. Each plot visually represents the nature of the quadratic form associated with matrices \( A_1, A_2, A_3, \) and \( A_4 \). Analyzing these can help understand the properties such as definiteness of the matrices.
Transcribed Image Text:# Matrix-Function Relationships ## Consider the Matrices \[ A_1 = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix}, \quad A_2 = \begin{pmatrix} -3 & 0 \\ 0 & -2 \end{pmatrix}, \quad A_3 = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}, \quad \text{and} \quad A_4 = \begin{pmatrix} -5 & -4 \\ -4 & 0 \end{pmatrix}. \] These matrices can be used to build functions \( x^T A_1 x, x^T A_2 x, x^T A_3 x, \) and \( x^T A_4 x \). These functions are represented (in an arbitrary order) in Figure 1. Match each matrix with its corresponding picture. ### Figure 1: Graphical Representation of Functions - **(a)** A surface plot showing a hyperbolic paraboloid shape spanning a positive and negative space, often associated with matrices having both positive and negative eigenvalues. - **(b)** A surface plot depicting a relatively flat plane that descends uniformly, typically representative of negative definite matrices. - **(c)** A surface plot with a dome-like upward curvature, usually indicating a positive definite matrix. - **(d)** A surface plot showing a saddle shape, indicative of an indefinite matrix with both positive and negative eigenvalues. In summary, these graphs illustrate the functions derived from the quadratic forms \( x^T A x \) for the given matrices. Each plot visually represents the nature of the quadratic form associated with matrices \( A_1, A_2, A_3, \) and \( A_4 \). Analyzing these can help understand the properties such as definiteness of the matrices.
Expert Solution
Step 1: Introduction of the given problem

We have to match the given matrix with the function represented in the graph 

steps

Step by step

Solved in 3 steps with 22 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS