Problem 0.6 Given a symmetric matrix A = for A = 69, (3) Diagonalize the matrix QA(x,y) = [x = y] [86] [*] (1) Check by a direct computation that Q₁(x, y) = ax² + 2bxy + cy². (2) Explain what {(x, y) = R² : Q₁(x, y) = 1} represents geometrically, and B = [8] 40 [2], respectively. [31 3 define Can you find an orthogonal matrix V such that VBV-¹ becomes diagonal? (4) Sketch the set {(x, y) = R² : 3x² + 2xy + 3y² = 1}, by relating it with QB, and using the ingredients from (1)-(3).
Problem 0.6 Given a symmetric matrix A = for A = 69, (3) Diagonalize the matrix QA(x,y) = [x = y] [86] [*] (1) Check by a direct computation that Q₁(x, y) = ax² + 2bxy + cy². (2) Explain what {(x, y) = R² : Q₁(x, y) = 1} represents geometrically, and B = [8] 40 [2], respectively. [31 3 define Can you find an orthogonal matrix V such that VBV-¹ becomes diagonal? (4) Sketch the set {(x, y) = R² : 3x² + 2xy + 3y² = 1}, by relating it with QB, and using the ingredients from (1)-(3).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 0.6:**
Given a symmetric matrix \( A = \begin{bmatrix} a & b \\ b & c \end{bmatrix} \), define
\[ Q_A(x, y) = \begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix} a & b \\ b & c \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}. \]
1. Check by a direct computation that \( Q_A(x, y) = ax^2 + 2bxy + cy^2 \).
2. Explain what \( \{(x, y) \in \mathbb{R}^2 : Q_A(x, y) = 1\} \) represents geometrically, for \( A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \), and \( \begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix} \), respectively.
3. Diagonalize the matrix
\[ B = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}. \]
Can you find an orthogonal matrix \( V \) such that \( VBV^{-1} \) becomes diagonal?
4. Sketch the set
\[ \{(x, y) \in \mathbb{R}^2 : 3x^2 + 2xy + 3y^2 = 1\}, \]
by relating it with \( Q_B \), and using the ingredients from (1)-(3).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ef06bb3-2d9b-4f27-bb3b-835b443ab608%2Fa56b263f-f510-42cd-80a8-a304b0b5c030%2F2xcqzml_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 0.6:**
Given a symmetric matrix \( A = \begin{bmatrix} a & b \\ b & c \end{bmatrix} \), define
\[ Q_A(x, y) = \begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix} a & b \\ b & c \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}. \]
1. Check by a direct computation that \( Q_A(x, y) = ax^2 + 2bxy + cy^2 \).
2. Explain what \( \{(x, y) \in \mathbb{R}^2 : Q_A(x, y) = 1\} \) represents geometrically, for \( A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \), and \( \begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix} \), respectively.
3. Diagonalize the matrix
\[ B = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}. \]
Can you find an orthogonal matrix \( V \) such that \( VBV^{-1} \) becomes diagonal?
4. Sketch the set
\[ \{(x, y) \in \mathbb{R}^2 : 3x^2 + 2xy + 3y^2 = 1\}, \]
by relating it with \( Q_B \), and using the ingredients from (1)-(3).
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