The columns of Q were obtained by applying the Gram-Schmidt process to the columns of A. Find an upper triangular matrix R such that A = QR. 5 404 A= Q= 2-2 -4 -3 2|7 22 5 7 22 N 4 7 2 7 22 A 1 22
The columns of Q were obtained by applying the Gram-Schmidt process to the columns of A. Find an upper triangular matrix R such that A = QR. 5 404 A= Q= 2-2 -4 -3 2|7 22 5 7 22 N 4 7 2 7 22 A 1 22
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
![**Problem Description:**
The columns of \( Q \) were obtained by applying the Gram-Schmidt process to the columns of \( A \). Find an upper triangular matrix \( R \) such that \( A = QR \).
**Matrix Representations:**
\[A = \begin{bmatrix}
2 & 3 \\
5 & 7 \\
2 & -2 \\
-4 & -3
\end{bmatrix}, \quad Q = \begin{bmatrix}
\frac{2}{7} & \frac{1}{\sqrt{22}} \\
\frac{5}{7} & \frac{7}{\sqrt{22}} \\
\frac{2}{7} & -\frac{4}{\sqrt{22}} \\
-\frac{4}{7} & \frac{1}{\sqrt{22}}
\end{bmatrix} \]
Below the matrices, there is a placeholder indicating where the user should input the matrix \( R \):
\[ R = \underline{\hspace{0.25in}} \]
### Detailed Explanation
**Gram-Schmidt Process:**
The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space. The outcome is that the matrix \( Q \) derived during this process contains orthonormal columns.
**Objective:**
Given matrices \( A \) and \( Q \), our objective is to determine the upper triangular matrix \( R \) such that the equation \( A = QR \) holds true.
**Steps to Solve:**
1. **Input the Matrices \( A \) and \( Q \):**
- Matrix \( A \) is a 4x2 matrix.
- Matrix \( Q \) is a 4x2 orthonormal matrix computed using the Gram-Schmidt process.
2. **Define the Matrix \( R \):**
- Since \( Q \) is orthonormal, \( R \) will be an upper triangular matrix.
- To find \( R \), solve \( R = Q^T A \) because \( Q \) is orthogonal (\( Q^T Q = I \)).
**Example Calculations:**
If \( Q \) and \( A \) are as defined above, calculate each element of \( R \) by performing the matrix](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5246d2a-5796-4784-a543-e596d3b5542c%2Fb0c1020d-5f4a-4a8a-b284-7b015a03ddc3%2Fhnnhvl_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
The columns of \( Q \) were obtained by applying the Gram-Schmidt process to the columns of \( A \). Find an upper triangular matrix \( R \) such that \( A = QR \).
**Matrix Representations:**
\[A = \begin{bmatrix}
2 & 3 \\
5 & 7 \\
2 & -2 \\
-4 & -3
\end{bmatrix}, \quad Q = \begin{bmatrix}
\frac{2}{7} & \frac{1}{\sqrt{22}} \\
\frac{5}{7} & \frac{7}{\sqrt{22}} \\
\frac{2}{7} & -\frac{4}{\sqrt{22}} \\
-\frac{4}{7} & \frac{1}{\sqrt{22}}
\end{bmatrix} \]
Below the matrices, there is a placeholder indicating where the user should input the matrix \( R \):
\[ R = \underline{\hspace{0.25in}} \]
### Detailed Explanation
**Gram-Schmidt Process:**
The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space. The outcome is that the matrix \( Q \) derived during this process contains orthonormal columns.
**Objective:**
Given matrices \( A \) and \( Q \), our objective is to determine the upper triangular matrix \( R \) such that the equation \( A = QR \) holds true.
**Steps to Solve:**
1. **Input the Matrices \( A \) and \( Q \):**
- Matrix \( A \) is a 4x2 matrix.
- Matrix \( Q \) is a 4x2 orthonormal matrix computed using the Gram-Schmidt process.
2. **Define the Matrix \( R \):**
- Since \( Q \) is orthonormal, \( R \) will be an upper triangular matrix.
- To find \( R \), solve \( R = Q^T A \) because \( Q \) is orthogonal (\( Q^T Q = I \)).
**Example Calculations:**
If \( Q \) and \( A \) are as defined above, calculate each element of \( R \) by performing the matrix
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 8 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
