IGNOU MMT 2 2026 - Find the QR decomposition of the matrix

Question

Find the QR decomposition of the matrix

$$\begin{bmatrix} 2 & -2 & 1 \\ 2 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}$

10 Jan 2026

Answer :

Word Count : 536

हमें दी गई मैट्रिक्स का QR डीकंपोज़िशन करना है: [ A = \begin{bmatrix} 2 & -2 & 1 \ 2 & 2 & 1 \ 0 & 1 & 1 \ 1 & 0 & 1 \end{bmatrix}. ] QR decomposition के लिए Gram-Schmidt प्रक्रिया अपनाते हैं। Step 1: कॉलम वेक्टर चुनें [ a_1 = \begin{bmatrix} 2 \ 2 \ 0 \ 1 \end{bmatrix}, \quad a_2 = \begin{bmatrix} -2 \ 2 \ 1 \ 0 \end{bmatrix}, \quad a_3 = \begin{bmatrix} 1 \ 1 \ 1 \ 1 \end{bmatrix}. ] Step 2: (q_1) निकालें [ q_1 = \frac{a_1}{|a_1|}. ] (|a_1| = \sqrt{2^2 + 2^2 + 0^2 + 1^2} = \sqrt{4 + 4 + 0 + 1} = \sqrt{9} = 3). [ q_1 = \frac{1}{3} \begin{bmatrix} 2 \ 2 \ 0 \ 1 \end{bmatrix} = \begin{bmatrix} 2/3 \ 2/3 \ 0 \ 1/3 \end{bmatrix}. ] Step 3: (q_2) निकालें [ u_2 = a_2 - \text{proj}*{q_1}(a_2), \quad \text{जहां } \text{proj}*{q_1}(a_2) = (q_1^T a_2) q_1 ] पहले, (q_1^T a_2 = \frac{2}{3}(-2) + \frac{2}{3}(2) + 0*1 + \frac{1}{3}(0) = -4/3 + 4/3 + 0 + 0 = 0). तो (\text{proj}_{q_1}(a_2) = 0 \cdot q_1 = 0). इसलिए (u_2 = a_2 = \begin{bmatrix}-2\2\1\0\end{bmatrix}). अब (q_2 = \frac{u_2}{|u_2|}) (|u_2| = \sqrt{(-2)^2 + 2^2 + 1^2 + 0^2} = \sqrt{4+4+1+0} = \sqrt{9} =

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