Math Problem Statement
Q = 1 3 2 −1 2 2 2 −1 −1 2 2 a) Show that Q is orthogonal.
Solution
To show that the matrix is orthogonal, we need to prove that , where is the transpose of , and is the identity matrix.
Given matrix :
\begin{pmatrix} 2 & -1 & 2 \\ 2 & 2 & -1 \\ -1 & 2 & 2 \end{pmatrix}$$ ### Step 1: Compute $$Q^T$$ The transpose $$Q^T$$ is obtained by switching rows with columns: $$Q^T = \frac{1}{\sqrt{3}} \begin{pmatrix} 2 & 2 & -1 \\ -1 & 2 & 2 \\ 2 & -1 & 2 \end{pmatrix}$$ ### Step 2: Compute $$Q^T Q$$ Multiply $$Q^T$$ by $$Q$$: $$Q^T Q = \frac{1}{\sqrt{3}} \begin{pmatrix} 2 & 2 & -1 \\ -1 & 2 & 2 \\ 2 & -1 & 2 \end{pmatrix} \cdot \frac{1}{\sqrt{3}} \begin{pmatrix} 2 & -1 & 2 \\ 2 & 2 & -1 \\ -1 & 2 & 2 \end{pmatrix}$$ The product becomes: $$Q^T Q = \frac{1}{3} \begin{pmatrix} (2)(2) + (2)(2) + (-1)(-1) & (2)(-1) + (2)(2) + (-1)(2) & (2)(2) + (2)(-1) + (-1)(2) \\ (-1)(2) + (2)(-1) + (2)(2) & (-1)(-1) + (2)(2) + (2)(2) & (-1)(2) + (2)(2) + (2)(-1) \\ (2)(2) + (-1)(2) + (2)(-1) & (2)(-1) + (-1)(2) + (2)(2) & (2)(2) + (-1)(-1) + (2)(2) \end{pmatrix}$$ Perform each calculation: 1. $$Q^T Q[1,1] = 4 + 4 + 1 = 9$$ 2. $$Q^T Q[1,2] = -2 + 4 - 2 = 0$$ 3.Continuing the calculations for $$Q^T Q$$: ### Step 3: Complete the Calculations 1. **Row 1, Column 1:** $$Q^T Q[1,1] = 4 + 4 + 1 = 9$$ 2. **Row 1, Column 2:** $$Q^T Q[1,2] = (-2) + 4 + (-2) = 0$$ 3. **Row 1, Column 3:** $$Q^T Q[1,3] = (4) + (-2) + (-2) = 0$$ 4. **Row 2, Column 1:** $$Q^T Q[2,1] = (-2) + (-2) + (4) = 0$$ 5. **Row 2, Column 2:** $$Q^T Q[2,2] = (1) + (4) + (4) = 9$$ 6. **Row 2, Column 3:** $$Q^T Q[2,3] = (-2) + (4) + (-2) = 0$$ 7. **Row 3, Column 1:** $$Q^T Q[3,1] = (4) + (-2) + (-2) = 0$$ 8. **Row 3, Column 2:** $$Q^T Q[3,2] = (-2) + (4) + (-2) = 0$$ 9. **Row 3, Column 3:** $$Q^T Q[3,3] = (4) + (1) + (4) = 9$$ ### Step 4: Normalize and Verify Identity Since each element of $$Q^T Q$$ is scaled by $$\frac{1}{3}$$, the final matrix is: $$Q^T Q = \frac{1}{3} \begin{pmatrix} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = I$$ ### Conclusion: Since $$Q^T Q = I$$, the matrix $$Q$$ is orthogonal. Would you like a breakdown of any particular step? Here are 5 questions for further exploration: 1. How do you compute the transpose of a matrix in general? 2. Why is the condition $$Q^T Q = I$$ sufficient to prove orthogonality? 3. Can a non-square matrix be orthogonal? Why or why not? 4. What is the geometric significance of an orthogonal matrix? 5. How would rounding errors affect verifying orthogonality in numerical computations? **Tip:** Always double-check each step when verifying matrix properties like orthogonality, as a small error can propagate through the calculations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Transposition
Matrix Multiplication
Orthogonal Matrices
Formulas
Orthogonality condition: Q^T Q = I
Matrix transpose: (A^T)[i, j] = A[j, i]
Matrix multiplication: C[i, j] = Σ(A[i, k] * B[k, j])
Theorems
Orthogonal Matrix Theorem
Suitable Grade Level
Grades 11-12, Undergraduate Linear Algebra
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