Math Problem Statement
Let A = [3 5 4 2] a (2x2) matrix. What is the maximum value of f(x) = x^TAx subject to the constraint x^Tx = 1 ?
Solution
The problem is to maximize subject to the constraint , where is a matrix given by:
3 & 5 \\ 4 & 2 \end{bmatrix}$$ This is a standard quadratic form optimization problem with a norm constraint. The method to solve this is to diagonalize $$A$$ by finding its eigenvalues and eigenvectors. The maximum value of $$f(x)$$ will correspond to the largest eigenvalue of $$A$$. ### Step 1: Eigenvalues of matrix $$A$$ The eigenvalues $$\lambda$$ are solutions to the characteristic equation $$\det(A - \lambda I) = 0$$. $$\det \left( \begin{bmatrix} 3 & 5 \\ 4 & 2 \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \right) = \det \begin{bmatrix} 3 - \lambda & 5 \\ 4 & 2 - \lambda \end{bmatrix} = 0$$ $$(3 - \lambda)(2 - \lambda) - (5)(4) = 0$$ Expanding this: $$(3 - \lambda)(2 - \lambda) = 6 - 5\lambda + \lambda^2$$ Thus: $$6 - 5\lambda + \lambda^2 - 20 = 0$$ Simplifying: $$\lambda^2 - 5\lambda - 14 = 0$$ ### Step 2: Solve for $$\lambda$$ The eigenvalues are found by solving the quadratic equation: $$\lambda = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)}$$ $$\lambda = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2}$$ $$\lambda = \frac{5 \pm 9}{2}$$ So the eigenvalues are: $$\lambda_1 = \frac{5 + 9}{2} = 7, \quad \lambda_2 = \frac{5 - 9}{2} = -2$$ ### Step 3: Maximum value of $$f(x)$$ The maximum value of $$f(x) = x^T A x$$ subject to $$x^T x = 1$$ is the largest eigenvalue of $$A$$, which is $$\lambda_1 = 7$$. Thus, the maximum value of $$f(x)$$ is **7**. Would you like more details or explanations on any step? Here are some related questions for practice: 1. How do you compute eigenvectors of a matrix? 2. What is the geometric interpretation of the quadratic form $$x^T A x$$? 3. Can you explain the Rayleigh quotient and how it relates to this problem? 4. What is the minimum value of $$f(x) = x^T A x$$ given the same constraint? 5. How can we generalize this approach to larger matrices? **Tip**: When optimizing quadratic forms with constraints, eigenvalues often provide direct answers, especially in symmetric matrices!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Quadratic Forms
Eigenvalues and Eigenvectors
Matrix Diagonalization
Formulas
f(x) = x^T A x
x^T x = 1
det(A - λI) = 0
Quadratic equation formula: λ = (-b ± √(b² - 4ac)) / 2a
Theorems
Eigenvalue Theorem
Spectral Theorem for symmetric matrices
Suitable Grade Level
University (Undergraduate Level)
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