Given the real symmetric matrix A: 1 2 -1 A = 2 8 -4 -1 -4 7 (a) Apply elementary row operation III three times to A to get upper triangular matrix U. (Do NOT use any other row ops.) (b) Construct a unit lower triangular matrix L that will undo those row ops, so that A =LU.
Given the real symmetric matrix A: 1 2 -1 A = 2 8 -4 -1 -4 7 (a) Apply elementary row operation III three times to A to get upper triangular matrix U. (Do NOT use any other row ops.) (b) Construct a unit lower triangular matrix L that will undo those row ops, so that A =LU.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.5: Iterative Methods For Computing Eigenvalues
Problem 51EQ
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![### Matrix Decomposition and Properties of Symmetric Matrices
#### Given the real symmetric matrix \( A \):
\[ A = \begin{pmatrix}
1 & 2 & -1 \\
2 & 8 & -4 \\
-1 & -4 & 7
\end{pmatrix} \]
1. **Elementary Row Operations:**
- **Step (a):** Apply elementary row operation III three times to matrix \( A \) to obtain an upper triangular matrix \( U \). (Note: Do **NOT** use any other types of row operations.)
2. **Unit Lower Triangular Matrix Construction:**
- **Step (b):** Construct a unit lower triangular matrix \( L \) that will undo those row operations applied in step (a), so that \( A = L U \).
3. **Diagonal Matrix Representation:**
- **Step (c):** Demonstrate that \( U \) can be represented as \( D L^T \) where \( D \) is a diagonal matrix with the same diagonal elements as \( U \).
4. **Matrix Computation:**
- **Step (d):** Find \( L_1 = L D^{1/2} \), and prove that \( A = L_1 L_1^T \).
5. **Mathematical Term Identification:**
- **Step (e):** Identify the correct mathematical term for the factorization of \( A \) as \( L_1 L_1^T \).
6. **Symmetric Matrix Type Determination:**
- **Step (f):** Determine which of the five types of real symmetric matrices \( A \) is. (Do **not** attempt to find the eigenvalues.)
### Explanation of Steps
1. **Elementary Row Operations:**
- Perform specific row operations to transform \( A \) into an upper triangular form. For example:
- Operation III: Row operations involving the interchange of rows based on certain rules.
2. **Unit Lower Triangular Matrix \( L \):**
- \( L \) is formed to reverse the specific row operations used in forming \( U \), maintaining property \( A = L U \).
3. **Diagonal Matrix \( D \):**
- Display how \( U \) can be expressed as \( D L^T \), where \( D \) captures the diagonal elements from](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7701ebcd-ae79-49a5-aa56-eff1f72328a6%2Fb96322ff-7545-4858-86c3-b2fb94a0a466%2Fwhrrlfl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix Decomposition and Properties of Symmetric Matrices
#### Given the real symmetric matrix \( A \):
\[ A = \begin{pmatrix}
1 & 2 & -1 \\
2 & 8 & -4 \\
-1 & -4 & 7
\end{pmatrix} \]
1. **Elementary Row Operations:**
- **Step (a):** Apply elementary row operation III three times to matrix \( A \) to obtain an upper triangular matrix \( U \). (Note: Do **NOT** use any other types of row operations.)
2. **Unit Lower Triangular Matrix Construction:**
- **Step (b):** Construct a unit lower triangular matrix \( L \) that will undo those row operations applied in step (a), so that \( A = L U \).
3. **Diagonal Matrix Representation:**
- **Step (c):** Demonstrate that \( U \) can be represented as \( D L^T \) where \( D \) is a diagonal matrix with the same diagonal elements as \( U \).
4. **Matrix Computation:**
- **Step (d):** Find \( L_1 = L D^{1/2} \), and prove that \( A = L_1 L_1^T \).
5. **Mathematical Term Identification:**
- **Step (e):** Identify the correct mathematical term for the factorization of \( A \) as \( L_1 L_1^T \).
6. **Symmetric Matrix Type Determination:**
- **Step (f):** Determine which of the five types of real symmetric matrices \( A \) is. (Do **not** attempt to find the eigenvalues.)
### Explanation of Steps
1. **Elementary Row Operations:**
- Perform specific row operations to transform \( A \) into an upper triangular form. For example:
- Operation III: Row operations involving the interchange of rows based on certain rules.
2. **Unit Lower Triangular Matrix \( L \):**
- \( L \) is formed to reverse the specific row operations used in forming \( U \), maintaining property \( A = L U \).
3. **Diagonal Matrix \( D \):**
- Display how \( U \) can be expressed as \( D L^T \), where \( D \) captures the diagonal elements from
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