Read the SageMath code for understanding, and evaluate the SageCell. Use your findings to provide a matrix H such that HAH¹ = C. A = -17 19 16 3 -5 -19 -3 20; C = 15 4083 31 81100 31 30576 31 2661 62 3262 31 11087 31 1597 62 2439 31 7376 31
Read the SageMath code for understanding, and evaluate the SageCell. Use your findings to provide a matrix H such that HAH¹ = C. A = -17 19 16 3 -5 -19 -3 20; C = 15 4083 31 81100 31 30576 31 2661 62 3262 31 11087 31 1597 62 2439 31 7376 31
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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![**Understanding SageMath Code for Matrix Transformation**
This educational content explores how to use SageMath to find a matrix \( H \) such that \( HAH^{-1} = C \).
**Given Matrices:**
\[ A = \begin{bmatrix} -17 & 19 & -3 \\ 16 & 3 & 20 \\ -5 & -19 & 15 \end{bmatrix} \]
\[ C = \begin{bmatrix} \frac{4083}{31} & \frac{2661}{62} & \frac{1597}{62} \\ \frac{8110}{31} & -\frac{3262}{31} & \frac{2439}{31} \\ \frac{30576}{31} & -\frac{11087}{31} & \frac{7376}{31} \end{bmatrix} \]
**SageMath Code Explanation:**
1. **Variables Initialization:**
```
var('a,b,c,d,f,g,h,j,k')
```
This line declares the variables which will represent the unknown elements of matrix \( H \).
2. **Matrix Definitions:**
```
A = matrix(3,3,[-17,19,-3,16,3,20,-5,-19,15])
C = matrix(3,3,[4083/31,2661/62,-1597/62,8110/31,-3262/31,2439/31,30576/31,-11087/31,7376/31])
```
Here, matrices \( A \) and \( C \) are defined using the specified elements.
3. **Unknown Matrix \( H \):**
```
H = matrix(3,3,[a,b,c,d,f,g,h,j,k])
```
The unknown matrix \( H \) is initialized as a 3x3 matrix containing the variables.
4. **Matrix Display:**
```
print(A); print(); print(C); print()
```
This section prints matrices \( A \) and \( C \).
5. **Solving for Matrix \( H \):**
```
print(solve((H*A-C*H).list(),[a,b,c,d,f,g,h,j,k]))
```
The solve function is used to find the values of the variables that satisfy the equation \( HAH](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0aa34ad3-f98c-40e8-8209-2950de228720%2Fbea859f1-b226-4ca7-a049-413d154fe392%2Ff5ueiam_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding SageMath Code for Matrix Transformation**
This educational content explores how to use SageMath to find a matrix \( H \) such that \( HAH^{-1} = C \).
**Given Matrices:**
\[ A = \begin{bmatrix} -17 & 19 & -3 \\ 16 & 3 & 20 \\ -5 & -19 & 15 \end{bmatrix} \]
\[ C = \begin{bmatrix} \frac{4083}{31} & \frac{2661}{62} & \frac{1597}{62} \\ \frac{8110}{31} & -\frac{3262}{31} & \frac{2439}{31} \\ \frac{30576}{31} & -\frac{11087}{31} & \frac{7376}{31} \end{bmatrix} \]
**SageMath Code Explanation:**
1. **Variables Initialization:**
```
var('a,b,c,d,f,g,h,j,k')
```
This line declares the variables which will represent the unknown elements of matrix \( H \).
2. **Matrix Definitions:**
```
A = matrix(3,3,[-17,19,-3,16,3,20,-5,-19,15])
C = matrix(3,3,[4083/31,2661/62,-1597/62,8110/31,-3262/31,2439/31,30576/31,-11087/31,7376/31])
```
Here, matrices \( A \) and \( C \) are defined using the specified elements.
3. **Unknown Matrix \( H \):**
```
H = matrix(3,3,[a,b,c,d,f,g,h,j,k])
```
The unknown matrix \( H \) is initialized as a 3x3 matrix containing the variables.
4. **Matrix Display:**
```
print(A); print(); print(C); print()
```
This section prints matrices \( A \) and \( C \).
5. **Solving for Matrix \( H \):**
```
print(solve((H*A-C*H).list(),[a,b,c,d,f,g,h,j,k]))
```
The solve function is used to find the values of the variables that satisfy the equation \( HAH
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