Using Python, to help solve the following problem. Provide an explanation of your solutions to the problem. 4. A symmetric matrix D is positive definite if xTTDx > 0 for any nonzero vector x. It can be proved that any symmetric, positive definite matrix D can be factored in the form D = LLT for some lower triangular matrix L with nonzero diagonal elements. This is called the Cholesky factorization of D. Consider the matrix [2.25 -3 4.5 A = -3 5 -10 4.5 -10 34 a. Is A positive definite? Explain. b. Find a lower triangular matrix L such that LLT = A.
Using Python, to help solve the following problem. Provide an explanation of your solutions to the problem. 4. A symmetric matrix D is positive definite if xTTDx > 0 for any nonzero vector x. It can be proved that any symmetric, positive definite matrix D can be factored in the form D = LLT for some lower triangular matrix L with nonzero diagonal elements. This is called the Cholesky factorization of D. Consider the matrix [2.25 -3 4.5 A = -3 5 -10 4.5 -10 34 a. Is A positive definite? Explain. b. Find a lower triangular matrix L such that LLT = A.
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![## Using Python to Solve Mathematical Problems: Cholesky Factorization
### Problem Statement
Using Python, to help solve the following problem. Provide an explanation of your solutions to the problem.
4. A symmetric matrix \( D \) is **positive definite** if \( x^T D x > 0 \) for any nonzero vector \( x \). It can be proved that any symmetric, positive definite matrix \( D \) can be factored in the form \( D = LL^T \) for some lower triangular matrix \( L \) with nonzero diagonal elements. This is called the **Cholesky factorization** of \( D \). Consider the matrix
\[
A = \begin{bmatrix}
2.25 & -3 & 4.5 \\
-3 & 5 & -10 \\
4.5 & -10 & 34
\end{bmatrix}.
\]
#### a. Is \( A \) positive definite? Explain.
#### b. Find a lower triangular matrix \( L \) such that \( LL^T = A \).
---
### Explanation of Concepts
**Positive Definite Matrix:** A symmetric matrix \( D \) is positive definite if \( x^T D x \) is greater than zero for any nonzero vector \( x \). This property ensures that all eigenvalues of the matrix are positive.
**Cholesky Factorization:** For any symmetric, positive definite matrix \( D \), there exists a unique lower triangular matrix \( L \) such that \( D = LL^T \). This \( L \) has nonzero diagonal elements, and the factorization simplifies various matrix computations.
### Steps to Solve the Problem
1. **Check Positive Definiteness of \( A \):**
- Verify if \( A \) is symmetric.
- Check the leading principal minors (determinants of the top-left \( k \times k \) submatrices) are all positive.
2. **Compute the Cholesky Factor \( L \):**
- Use an algorithm or a computational tool like Python to find the lower triangular matrix \( L \) such that \( LL^T = A \).
### Practical Application Using Python
Here's an implementation in Python to address the problem:
```python
import numpy as np
# Define matrix A
A = np.array([[2.25, -3, 4.5],
[-3, 5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe6e77107-ef1d-4e39-badf-4decafa47f1b%2F9cf50a2e-d428-4402-beca-94ba187feadd%2Ful4qhp_processed.png&w=3840&q=75)
Transcribed Image Text:## Using Python to Solve Mathematical Problems: Cholesky Factorization
### Problem Statement
Using Python, to help solve the following problem. Provide an explanation of your solutions to the problem.
4. A symmetric matrix \( D \) is **positive definite** if \( x^T D x > 0 \) for any nonzero vector \( x \). It can be proved that any symmetric, positive definite matrix \( D \) can be factored in the form \( D = LL^T \) for some lower triangular matrix \( L \) with nonzero diagonal elements. This is called the **Cholesky factorization** of \( D \). Consider the matrix
\[
A = \begin{bmatrix}
2.25 & -3 & 4.5 \\
-3 & 5 & -10 \\
4.5 & -10 & 34
\end{bmatrix}.
\]
#### a. Is \( A \) positive definite? Explain.
#### b. Find a lower triangular matrix \( L \) such that \( LL^T = A \).
---
### Explanation of Concepts
**Positive Definite Matrix:** A symmetric matrix \( D \) is positive definite if \( x^T D x \) is greater than zero for any nonzero vector \( x \). This property ensures that all eigenvalues of the matrix are positive.
**Cholesky Factorization:** For any symmetric, positive definite matrix \( D \), there exists a unique lower triangular matrix \( L \) such that \( D = LL^T \). This \( L \) has nonzero diagonal elements, and the factorization simplifies various matrix computations.
### Steps to Solve the Problem
1. **Check Positive Definiteness of \( A \):**
- Verify if \( A \) is symmetric.
- Check the leading principal minors (determinants of the top-left \( k \times k \) submatrices) are all positive.
2. **Compute the Cholesky Factor \( L \):**
- Use an algorithm or a computational tool like Python to find the lower triangular matrix \( L \) such that \( LL^T = A \).
### Practical Application Using Python
Here's an implementation in Python to address the problem:
```python
import numpy as np
# Define matrix A
A = np.array([[2.25, -3, 4.5],
[-3, 5
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