Find the standard Matrix Find the standard matrix for the following Q(x) quadratic form Find the eigenvalues Calculate the eigenvalues for the above standard matrix Determine the larger eigenvalue: 19 Determine the second eigenvalue: e Determine the eigenvectors Next, we determine the corresponding eigenvectors for each eigenvalue. At this point we do not yet need to normalize our eigenvectors, but we will do that in a later step Pmatrix If the following vectors are the respective eigenvectors of A, determine the orthogonal matrix P such that A PDP™ With [2] as the first vector in P Hint: Quadratic form new equation Q(x) = 10z² + 122,2₂ +152 10 6 6 The first eigenvector for A=19 has the form (1) Assuming that z=2, enter the eigenvector for A = 19 = 2 3 4 The next eigenvector for A-6 has the form ty=1). Assuming that z=-3, enter the eigenvector for X = 6 A 3 2 2isgrt13 15 3/sqrt13 --3 -2
Find the standard Matrix Find the standard matrix for the following Q(x) quadratic form Find the eigenvalues Calculate the eigenvalues for the above standard matrix Determine the larger eigenvalue: 19 Determine the second eigenvalue: e Determine the eigenvectors Next, we determine the corresponding eigenvectors for each eigenvalue. At this point we do not yet need to normalize our eigenvectors, but we will do that in a later step Pmatrix If the following vectors are the respective eigenvectors of A, determine the orthogonal matrix P such that A PDP™ With [2] as the first vector in P Hint: Quadratic form new equation Q(x) = 10z² + 122,2₂ +152 10 6 6 The first eigenvector for A=19 has the form (1) Assuming that z=2, enter the eigenvector for A = 19 = 2 3 4 The next eigenvector for A-6 has the form ty=1). Assuming that z=-3, enter the eigenvector for X = 6 A 3 2 2isgrt13 15 3/sqrt13 --3 -2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Find the Standard Matrix
**Objective:**
Find the standard matrix for the quadratic form \( Q(\mathbf{x}) \).
**Quadratic Form:**
\[ Q(\mathbf{x}) = 10x_1^2 + 12x_1x_2 + 15x_2^2 \]
**Standard Matrix:**
\[
\begin{bmatrix}
10 & 6 \\
6 & 15
\end{bmatrix}
\]
---
### Find the Eigenvalues
**Objective:**
Calculate the eigenvalues for the above standard matrix.
- **Determine the larger eigenvalue:** 19
- **Determine the second eigenvalue:** 6
---
### Determine the Eigenvectors
**Objective:**
Find the corresponding eigenvectors for each eigenvalue.
- The first eigenvector for \( \lambda = 19 \) has the form \( \mathbf{v}_1 = \begin{bmatrix} x_1 \\ 2 \\ 3 \end{bmatrix} \). Assuming that \( x_1 = 2 \), enter the eigenvector for \( \lambda = 19 \):
\[
\begin{bmatrix}
2 \\
2 \\
3
\end{bmatrix}
\]
- The next eigenvector for \( \lambda = 6 \) has the form \( \mathbf{v}_2 = \begin{bmatrix} x_1 \\ 3 \\ 2 \end{bmatrix} \). Assuming that \( x_1 = -3 \), enter the eigenvector for \( \lambda = 6 \):
\[
\begin{bmatrix}
-3 \\
3 \\
2
\end{bmatrix}
\]
---
### P-matrix
**Objective:**
If the following vectors are the respective eigenvectors of \( A \), determine the orthogonal matrix \( P \) such that \( A = PDP^{T} \).
With \(\begin{bmatrix} \frac{2}{\sqrt{13}} \\ \frac{3}{\sqrt{13}} \end{bmatrix}\) as the first vector in \( P \):
\[
\begin{bmatrix}
\frac{2}{\sqrt{13}} & -3 \\
\frac{3}{\sqrt{13}} & 2
\end{bmatrix}
\]
**Hint:**
No specific](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b609ed5-0879-4e23-8eee-4515e84ef2b9%2F40c3a1e3-f403-4518-80ab-b38473434d99%2Fx8o5tuk_processed.png&w=3840&q=75)
Transcribed Image Text:### Find the Standard Matrix
**Objective:**
Find the standard matrix for the quadratic form \( Q(\mathbf{x}) \).
**Quadratic Form:**
\[ Q(\mathbf{x}) = 10x_1^2 + 12x_1x_2 + 15x_2^2 \]
**Standard Matrix:**
\[
\begin{bmatrix}
10 & 6 \\
6 & 15
\end{bmatrix}
\]
---
### Find the Eigenvalues
**Objective:**
Calculate the eigenvalues for the above standard matrix.
- **Determine the larger eigenvalue:** 19
- **Determine the second eigenvalue:** 6
---
### Determine the Eigenvectors
**Objective:**
Find the corresponding eigenvectors for each eigenvalue.
- The first eigenvector for \( \lambda = 19 \) has the form \( \mathbf{v}_1 = \begin{bmatrix} x_1 \\ 2 \\ 3 \end{bmatrix} \). Assuming that \( x_1 = 2 \), enter the eigenvector for \( \lambda = 19 \):
\[
\begin{bmatrix}
2 \\
2 \\
3
\end{bmatrix}
\]
- The next eigenvector for \( \lambda = 6 \) has the form \( \mathbf{v}_2 = \begin{bmatrix} x_1 \\ 3 \\ 2 \end{bmatrix} \). Assuming that \( x_1 = -3 \), enter the eigenvector for \( \lambda = 6 \):
\[
\begin{bmatrix}
-3 \\
3 \\
2
\end{bmatrix}
\]
---
### P-matrix
**Objective:**
If the following vectors are the respective eigenvectors of \( A \), determine the orthogonal matrix \( P \) such that \( A = PDP^{T} \).
With \(\begin{bmatrix} \frac{2}{\sqrt{13}} \\ \frac{3}{\sqrt{13}} \end{bmatrix}\) as the first vector in \( P \):
\[
\begin{bmatrix}
\frac{2}{\sqrt{13}} & -3 \\
\frac{3}{\sqrt{13}} & 2
\end{bmatrix}
\]
**Hint:**
No specific
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