Math Problem Statement
The columns of Q were obtained by applying the Gram-Schmidt Process to the columns of A. Find the upper triangular matrix R such that A = QR. A =
2 9 6
4 10 −7 −4 −2 2 , Q =
1 3
2 3
2 3
2 3
1 3 − 2 3 − 2 3
2 3 − 1 3
Solution
To find the upper triangular matrix such that , follow these steps:
Step 1: Define the matrices and
Given:
\begin{bmatrix} 2 & 9 & 6 \\ 4 & 10 & -7 \\ -4 & -2 & 2 \end{bmatrix}, \quad Q = \begin{bmatrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & -\frac{2}{3} \\ -\frac{2}{3} & \frac{2}{3} & -\frac{1}{3} \end{bmatrix}.$$ ### Step 2: Use the formula $$R = Q^\top A$$ The matrix $$R$$ is found by multiplying the transpose of $$Q$$ by $$A$$: $$R = Q^\top A.$$ ### Step 3: Compute $$Q^\top$$ The transpose of $$Q$$ is: $$Q^\top = \begin{bmatrix} \frac{1}{3} & \frac{2}{3} & -\frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \frac{2}{3} & -\frac{2}{3} & -\frac{1}{3} \end{bmatrix}.$$ ### Step 4: Multiply $$Q^\top$$ with $$A$$ Perform the matrix multiplication: $$R = Q^\top A = \begin{bmatrix} \frac{1}{3} & \frac{2}{3} & -\frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \frac{2}{3} & -\frac{2}{3} & -\frac{1}{3} \end{bmatrix} \begin{bmatrix} 2 & 9 & 6 \\ 4 & 10 & -7 \\ -4 & -2 & 2 \end{bmatrix}.$$ ### Step 5: Compute each entry of $$R$$ After performing the matrix multiplication: $$R = \begin{bmatrix} 6 & 9 & 6 \\ 0 & 6 & 3 \\ 0 & 0 & 3 \end{bmatrix}.$$ ### Final Answer The upper triangular matrix $$R$$ is: $$R = \begin{bmatrix} 6 & 9 & 6 \\ 0 & 6 & 3 \\ 0 & 0 & 3 \end{bmatrix}.$$ ### Verification You can verify $$A = QR$$ by multiplying $$Q$$ and $$R$$ back together. Do you want me to verify or explain any specific part further? --- #### Related Questions: 1. What is the Gram-Schmidt process and how is it applied to a matrix? 2. Why is $$R$$ upper triangular in the QR decomposition? 3. How can $$Q$$ be verified as an orthogonal matrix? 4. What are the properties of matrices $$Q$$ and $$R$$ in QR decomposition? 5. Can the QR decomposition be computed for non-square matrices? #### Tip: When verifying $$A = QR$$, always check that $$Q$$ is orthogonal and $$R$$ is upper triangular to ensure the decomposition is correct.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
QR Decomposition
Matrix Multiplication
Orthogonal Matrices
Formulas
R = Q^T * A
A = QR
Theorems
Gram-Schmidt Process
Properties of Orthogonal Matrices
Suitable Grade Level
Undergraduate Mathematics (Linear Algebra)