4. Consider the matrix 1 0 0 0 0 0 -2a -a (a² + a) 0 (2-a²) - (2-a²) -2a (2-a²) 2+ 4a - a² Find all values of a for which this matrix is psd. Are there values of a for which the matrix is pd?
4. Consider the matrix 1 0 0 0 0 0 -2a -a (a² + a) 0 (2-a²) - (2-a²) -2a (2-a²) 2+ 4a - a² Find all values of a for which this matrix is psd. Are there values of a for which the matrix is pd?
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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Question
Please answer question 4
![**Educational Text Transcription and Explanation**
---
1. **Consider the matrices**
\[
A_1 = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix},
A_2 = \begin{pmatrix} -3 & 0 \\ 0 & -2 \end{pmatrix},
A_3 = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix},
A_4 = \begin{pmatrix} -5 & -4 \\ -4 & 0 \end{pmatrix}.
\]
These matrices can be used to build functions \(x^TA_1x\), \(x^TA_2x\), \(x^TA_3x\), and \(x^TA_4x\). 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)**: [Graph of surface corresponding to one of the functions.]
- **(b)**: [Graph of surface corresponding to one of the functions.]
- **(c)**: [Graph of surface corresponding to one of the functions.]
- **(d)**: [Graph of surface corresponding to one of the functions.]
Explain briefly.
2. **Consider matrices of the form**
\[
M_{\alpha, \beta} := \alpha \begin{pmatrix} 1 & -2 \\ -2 & 2 \end{pmatrix} + \beta \begin{pmatrix} -1 & 1 \\ 1 & -2 \end{pmatrix}
\]
where \(\alpha\) and \(\beta\) are scalars. Depending on the values of \(\alpha\) and \(\beta\), these matrices might be positive semi-definite (psd), negative semi-definite (nsd), or indefinite. In a graph with axes \(\alpha\) and \(\beta\), represent the set of all \((\alpha, \beta)\) values for which the matrix:
(a) \( M_{\alpha, \beta} \geq 0\).
(b) \( M_{\alpha, \beta} \leq 0\).
3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec417cb7-f4c3-4376-ad3a-c1f5cc0351ef%2F0f648e93-e053-47e4-b929-b6f55bfcee65%2F35matp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Text Transcription and Explanation**
---
1. **Consider the matrices**
\[
A_1 = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix},
A_2 = \begin{pmatrix} -3 & 0 \\ 0 & -2 \end{pmatrix},
A_3 = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix},
A_4 = \begin{pmatrix} -5 & -4 \\ -4 & 0 \end{pmatrix}.
\]
These matrices can be used to build functions \(x^TA_1x\), \(x^TA_2x\), \(x^TA_3x\), and \(x^TA_4x\). 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)**: [Graph of surface corresponding to one of the functions.]
- **(b)**: [Graph of surface corresponding to one of the functions.]
- **(c)**: [Graph of surface corresponding to one of the functions.]
- **(d)**: [Graph of surface corresponding to one of the functions.]
Explain briefly.
2. **Consider matrices of the form**
\[
M_{\alpha, \beta} := \alpha \begin{pmatrix} 1 & -2 \\ -2 & 2 \end{pmatrix} + \beta \begin{pmatrix} -1 & 1 \\ 1 & -2 \end{pmatrix}
\]
where \(\alpha\) and \(\beta\) are scalars. Depending on the values of \(\alpha\) and \(\beta\), these matrices might be positive semi-definite (psd), negative semi-definite (nsd), or indefinite. In a graph with axes \(\alpha\) and \(\beta\), represent the set of all \((\alpha, \beta)\) values for which the matrix:
(a) \( M_{\alpha, \beta} \geq 0\).
(b) \( M_{\alpha, \beta} \leq 0\).
3
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