Use Numpy to complete the following task(s). 5. Solve Assignment 5.1 #1. 1. (Tucker) A growth model for elephants (E) and mice (M) predicts population changes from decade to decade. E +3E+M M+2E+ 4M a. Determine the eigenvalues and associated eigenvectors for this system.
Use Numpy to complete the following task(s). 5. Solve Assignment 5.1 #1. 1. (Tucker) A growth model for elephants (E) and mice (M) predicts population changes from decade to decade. E +3E+M M+2E+ 4M a. Determine the eigenvalues and associated eigenvectors for this system.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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![**NumPy Assignment: Population Growth Model**
### Task 5: Solve Assignment 5.1 #1 using NumPy
1. **(Tucker) Growth Model Analysis**
A growth model for elephants (\(E\)) and mice (\(M\)) predicts population changes from decade to decade as follows:
\[
\begin{align*}
E & \leftarrow 3E + M \\
M & \leftarrow 2E + 4M
\end{align*}
\]
a. **Determine the eigenvalues and associated eigenvectors for this system.**
### Explanation:
In the given system of equations, the growth model for elephants (\(E\)) and mice (\(M\)) is described by the following linear transformations:
- \(E \leftarrow 3E + M\)
- \(M \leftarrow 2E + 4M\)
To find the eigenvalues and eigenvectors, follow these steps:
1. **Express the system in matrix form:**
\[
\begin{pmatrix}
E' \\
M'
\end{pmatrix}
=
\begin{pmatrix}
3 & 1 \\
2 & 4
\end{pmatrix}
\begin{pmatrix}
E \\
M
\end{pmatrix}
\]
2. **Find the eigenvalues (\(\lambda\)) by solving the characteristic equation:**
\[
\det(A - \lambda I) = 0
\]
Where \(A\) is the coefficient matrix:
\[
A =
\begin{pmatrix}
3 & 1 \\
2 & 4
\end{pmatrix}
\]
And \(I\) is the identity matrix:
\[
I =
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\]
3. **Compute the eigenvectors for the corresponding eigenvalues to understand the dynamics of population changes over time.**
Use NumPy in a Python environment to perform these computations and derive the eigenvalues and eigenvectors numerically. This will help analyze the stability and long-term behavior of the population model for elephants and mice.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea6ca246-da4e-4258-b850-8b268fd070fb%2Fc423eb94-3322-4161-bae9-298c77d36585%2F1irjvjs_processed.png&w=3840&q=75)
Transcribed Image Text:**NumPy Assignment: Population Growth Model**
### Task 5: Solve Assignment 5.1 #1 using NumPy
1. **(Tucker) Growth Model Analysis**
A growth model for elephants (\(E\)) and mice (\(M\)) predicts population changes from decade to decade as follows:
\[
\begin{align*}
E & \leftarrow 3E + M \\
M & \leftarrow 2E + 4M
\end{align*}
\]
a. **Determine the eigenvalues and associated eigenvectors for this system.**
### Explanation:
In the given system of equations, the growth model for elephants (\(E\)) and mice (\(M\)) is described by the following linear transformations:
- \(E \leftarrow 3E + M\)
- \(M \leftarrow 2E + 4M\)
To find the eigenvalues and eigenvectors, follow these steps:
1. **Express the system in matrix form:**
\[
\begin{pmatrix}
E' \\
M'
\end{pmatrix}
=
\begin{pmatrix}
3 & 1 \\
2 & 4
\end{pmatrix}
\begin{pmatrix}
E \\
M
\end{pmatrix}
\]
2. **Find the eigenvalues (\(\lambda\)) by solving the characteristic equation:**
\[
\det(A - \lambda I) = 0
\]
Where \(A\) is the coefficient matrix:
\[
A =
\begin{pmatrix}
3 & 1 \\
2 & 4
\end{pmatrix}
\]
And \(I\) is the identity matrix:
\[
I =
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\]
3. **Compute the eigenvectors for the corresponding eigenvalues to understand the dynamics of population changes over time.**
Use NumPy in a Python environment to perform these computations and derive the eigenvalues and eigenvectors numerically. This will help analyze the stability and long-term behavior of the population model for elephants and mice.
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