Math Problem Statement
Use QR Decomposition to solve the least square problem for the inconsistent system Ax = B, where A = [[2, 0], [0, 1], [1, 2]] and B = [[1], [0], [3]].
Solution
We are tasked with solving the least squares problem using QR decomposition for the inconsistent system , where:
2 & 0 \\ 0 & 1 \\ 1 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}$$ ### Step-by-step approach: 1. **QR Decomposition of $$A$$:** We begin by decomposing $$A$$ into the product of an orthogonal matrix $$Q$$ and an upper triangular matrix $$R$$: $$A = QR$$ 2. **Solve the system $$Ax = B$$ using $$QR$$:** After the QR decomposition, solving $$A\mathbf{x} = \mathbf{B}$$ becomes: $$QR\mathbf{x} = \mathbf{B}$$ Multiplying both sides by $$Q^\top$$ (transpose of $$Q$$) gives: $$R\mathbf{x} = Q^\top \mathbf{B}$$ Since $$R$$ is upper triangular, this equation can be solved using back-substitution. I will now compute the QR decomposition and solve the system.The QR decomposition of matrix $$A$$ gives: $$Q = \begin{pmatrix} -0.8944 & 0.3904 \\ 0 & -0.4880 \\ -0.4472 & -0.7807 \end{pmatrix}, \quad R = \begin{pmatrix} -2.2361 & -0.8944 \\ 0 & -2.0494 \end{pmatrix}$$ The transformed system $$R\mathbf{x} = Q^\top \mathbf{B}$$ yields: $$Q^\top \mathbf{B} = \begin{pmatrix} -2.2361 \\ -1.9518 \end{pmatrix}$$ Solving this system using back substitution gives the solution for $$\mathbf{x}$$: $$\mathbf{x} = \begin{pmatrix} 0.6190 \\ 0.9524 \end{pmatrix}$$ Thus, the least squares solution to the system $$A\mathbf{x} = \mathbf{B}$$ is approximately: $$\mathbf{x} = \begin{pmatrix} 0.619 \\ 0.952 \end{pmatrix}$$ Would you like further details or have any questions? Here are 5 related questions: 1. How is QR decomposition generally used in solving least squares problems? 2. Why is the system considered inconsistent? 3. What are the properties of the matrix $$Q$$ from QR decomposition? 4. How would the solution change if matrix $$A$$ were overdetermined (more rows than columns)? 5. What is the geometric interpretation of a least squares solution? **Tip:** QR decomposition is more stable for numerical solutions than the normal equation approach when solving least squares problems.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
QR Decomposition
Least Squares
Formulas
QR Decomposition: A = QR
Least Squares Solution: R * x = Q^T * B
Theorems
QR Decomposition Theorem
Least Squares Approximation
Suitable Grade Level
Undergraduate
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