4 -8] -2 10 -8 -1 5 If the Gram-Schmidt process is applied to determine the QR factorization of A, then, after the first two orthomormal vectors q₁ and q₂ are computed, we have: [-2 Consider A = 33 33 ²33 1 3 2 333 4, R = 3-9 9 06 0 0 Finish the process; determine q3 and fill in the third column of Q and R. -6 6
4 -8] -2 10 -8 -1 5 If the Gram-Schmidt process is applied to determine the QR factorization of A, then, after the first two orthomormal vectors q₁ and q₂ are computed, we have: [-2 Consider A = 33 33 ²33 1 3 2 333 4, R = 3-9 9 06 0 0 Finish the process; determine q3 and fill in the third column of Q and R. -6 6
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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![Consider the matrix \( A \) given by:
\[ A = \begin{bmatrix} -2 & 4 & -8 \\ -2 & 10 & -8 \\ -1 & -1 & 5 \end{bmatrix} \]
If the Gram-Schmidt process is applied to determine the \( QR \) factorization of \( A \), then, after the first two orthonormal vectors \( \mathbf{q_1} \) and \( \mathbf{q_2} \) are computed, we have:
\[ Q = \begin{bmatrix}
- \frac{2}{3} & - \frac{1}{3} & \square \\
- \frac{2}{3} & 2 & \square \\
- \frac{1}{3} & -2 & \square
\end{bmatrix}, \quad R = \begin{bmatrix}
3 & -9 & 9 \\
0 & 6 & -6 \\
0 & 0 & 6
\end{bmatrix} \]
Finish the process; determine \( \mathbf{q_3} \) and fill in the third column of \( Q \) and \( R \).
Notice the partially completed orthonormal matrix \( Q \) and the upper triangular matrix \( R \). To complete the QR factorization:
1. \( Q \) should result in an orthogonal (orthonormal if vectors are normalized) matrix, with each column representing an orthonormal vector.
2. \( R \) should be an upper triangular matrix whose non-zero entries correspond to projections as determined by the Gram-Schmidt process.
For example:
- To complete the process determine the orthonormal vector \( \mathbf{q_3} \).
- Next, fill in the corresponding entries in the third column of both \( Q \) and \( R \).
The updated matrices \( Q \) and \( R \) will have their third columns completed as specified.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2F5bbf298b-1e48-4e51-bf8d-304bc1758ff8%2Fro108aj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the matrix \( A \) given by:
\[ A = \begin{bmatrix} -2 & 4 & -8 \\ -2 & 10 & -8 \\ -1 & -1 & 5 \end{bmatrix} \]
If the Gram-Schmidt process is applied to determine the \( QR \) factorization of \( A \), then, after the first two orthonormal vectors \( \mathbf{q_1} \) and \( \mathbf{q_2} \) are computed, we have:
\[ Q = \begin{bmatrix}
- \frac{2}{3} & - \frac{1}{3} & \square \\
- \frac{2}{3} & 2 & \square \\
- \frac{1}{3} & -2 & \square
\end{bmatrix}, \quad R = \begin{bmatrix}
3 & -9 & 9 \\
0 & 6 & -6 \\
0 & 0 & 6
\end{bmatrix} \]
Finish the process; determine \( \mathbf{q_3} \) and fill in the third column of \( Q \) and \( R \).
Notice the partially completed orthonormal matrix \( Q \) and the upper triangular matrix \( R \). To complete the QR factorization:
1. \( Q \) should result in an orthogonal (orthonormal if vectors are normalized) matrix, with each column representing an orthonormal vector.
2. \( R \) should be an upper triangular matrix whose non-zero entries correspond to projections as determined by the Gram-Schmidt process.
For example:
- To complete the process determine the orthonormal vector \( \mathbf{q_3} \).
- Next, fill in the corresponding entries in the third column of both \( Q \) and \( R \).
The updated matrices \( Q \) and \( R \) will have their third columns completed as specified.
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