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Question
Math
Posted 11 days ago

QR Factorisation
Find QRQ R factorisation of the following matrix:
A=(54126)
A=\left(\begin{array}{cc}
5 & -4 \\
12 & 6
\end{array}\right)


Answer:
\square
Note:
Enter your answer as two element Maple list: [Q,R][Q, R]. Each element of the list is Maple matrix: < col1 | col2 | . . . . Each col1, col2,... is matrix column in Maple vector syntax: a,b,\langle a, b, \ldots.

Note 2:
Double check your answer by multiplying QRQ R before submitting this form. Use preview function to ensure that your answer is typo-free.

Note 3:
if you need to type square root, use sqrt Maple function.
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Answer from Sia
Posted 11 days ago
Solution by Steps
step 1
Given matrix A=(54126) A = \begin{pmatrix} 5 & -4 \\ 12 & 6 \end{pmatrix} , we need to find its QR factorization
step 2
First, we find the orthonormal basis vectors for the columns of A A . Let a1=(512) a_1 = \begin{pmatrix} 5 \\ 12 \end{pmatrix} and a2=(46) a_2 = \begin{pmatrix} -4 \\ 6 \end{pmatrix}
step 3
Normalize a1 a_1 to get q1 q_1 : q1=a1a1=113(512)=(5131213) q_1 = \frac{a_1}{\|a_1\|} = \frac{1}{13} \begin{pmatrix} 5 \\ 12 \end{pmatrix} = \begin{pmatrix} \frac{5}{13} \\ \frac{12}{13} \end{pmatrix}
step 4
Project a2 a_2 onto q1 q_1 and subtract to get the orthogonal component: projq1(a2)=((5131213)(46))(5131213)=2013(5131213)=(100169240169) \text{proj}_{q_1}(a_2) = \left( \begin{pmatrix} \frac{5}{13} \\ \frac{12}{13} \end{pmatrix} \cdot \begin{pmatrix} -4 \\ 6 \end{pmatrix} \right) \begin{pmatrix} \frac{5}{13} \\ \frac{12}{13} \end{pmatrix} = \frac{20}{13} \begin{pmatrix} \frac{5}{13} \\ \frac{12}{13} \end{pmatrix} = \begin{pmatrix} \frac{100}{169} \\ \frac{240}{169} \end{pmatrix}
step 5
Subtract the projection from a2 a_2 to get u2 u_2 : u2=a2projq1(a2)=(46)(100169240169)=(776169774169) u_2 = a_2 - \text{proj}_{q_1}(a_2) = \begin{pmatrix} -4 \\ 6 \end{pmatrix} - \begin{pmatrix} \frac{100}{169} \\ \frac{240}{169} \end{pmatrix} = \begin{pmatrix} -\frac{776}{169} \\ \frac{774}{169} \end{pmatrix}
step 6
Normalize u2 u_2 to get q2 q_2 : q2=u2u2=1(776169)2+(774169)2(776169774169)=(1213513) q_2 = \frac{u_2}{\|u_2\|} = \frac{1}{\sqrt{\left( -\frac{776}{169} \right)^2 + \left( \frac{774}{169} \right)^2}} \begin{pmatrix} -\frac{776}{169} \\ \frac{774}{169} \end{pmatrix} = \begin{pmatrix} -\frac{12}{13} \\ \frac{5}{13} \end{pmatrix}
step 7
Form the orthogonal matrix Q Q : Q=(51312131213513) Q = \begin{pmatrix} \frac{5}{13} & -\frac{12}{13} \\ \frac{12}{13} & \frac{5}{13} \end{pmatrix}
step 8
Compute the upper triangular matrix R R using R=QTA R = Q^T A : R=(51312131213513)(54126)=(13406) R = \begin{pmatrix} \frac{5}{13} & \frac{12}{13} \\ -\frac{12}{13} & \frac{5}{13} \end{pmatrix} \begin{pmatrix} 5 & -4 \\ 12 & 6 \end{pmatrix} = \begin{pmatrix} 13 & 4 \\ 0 & 6 \end{pmatrix}
Answer
The QR factorization of the matrix A A is: Q=(51312131213513),R=(13406) Q = \begin{pmatrix} \frac{5}{13} & -\frac{12}{13} \\ \frac{12}{13} & \frac{5}{13} \end{pmatrix}, \quad R = \begin{pmatrix} 13 & 4 \\ 0 & 6 \end{pmatrix}
Key Concept
QR Factorization
Explanation
QR factorization decomposes a matrix into an orthogonal matrix Q Q and an upper triangular matrix R R . This is useful for solving linear systems and eigenvalue problems.

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