Math Problem Statement
Given the functionf:R2→Rdefined byf(x) =xᵀAx, wherexis a 2-dimensional column vector andA=(1 3−1 2). Determine the maximum and minimum values offwithinthe closed unit ball, i.e.,max‖x‖≤1f(x),min‖x‖≤1f(x).1
Solution
Let's analyze the function , where is the matrix:
1 & 3 \\ -1 & 2 \end{pmatrix}$$ ### 1. Understanding the Function: Given $$f(x) = x^T A x$$, where $$x$$ is a 2-dimensional column vector $$x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$, the function is a quadratic form. The goal is to find the maximum and minimum values of this quadratic form on the unit ball $$\|x\| \leq 1$$, i.e., the set of vectors $$x$$ such that $$x_1^2 + x_2^2 \leq 1$$. ### 2. Eigenvalues and Eigenvectors of $$A$$: To find the maximum and minimum of $$f(x)$$ on the unit ball, we need to study the properties of the matrix $$A$$. The extremal values of the quadratic form $$f(x)$$ are determined by the eigenvalues of $$A$$, and these correspond to the maximum and minimum values of $$f(x)$$ on the unit circle. We first need to compute the eigenvalues of $$A$$. The eigenvalues $$\lambda$$ are found by solving the characteristic equation: $$\det(A - \lambda I) = 0$$ This gives the equation: $$\det\begin{pmatrix} 1-\lambda & 3 \\ -1 & 2-\lambda \end{pmatrix} = 0$$ Expanding the determinant: $$(1-\lambda)(2-\lambda) - (-3) = 0$$ $$(1-\lambda)(2-\lambda) + 3 = 0$$ Expanding: $$2 - \lambda - 2\lambda + \lambda^2 + 3 = 0$$ $$\lambda^2 - 3\lambda + 5 = 0$$ The discriminant of this quadratic equation is: $$\Delta = (-3)^2 - 4 \cdot 1 \cdot 5 = 9 - 20 = -11$$ Since the discriminant is negative, the eigenvalues of $$A$$ are complex, which means the function $$f(x) = x^T A x$$ does not achieve real maxima or minima within the real domain. Therefore, we conclude that the function $$f(x)$$ does not have any real maximum or minimum values in the standard sense on the unit ball. ### 3. Conclusion: Given that the eigenvalues are complex and the function is defined in the real numbers, the maximum and minimum values of the function on the unit ball do not exist in real terms. Do you need any further explanation or clarification? Here are some follow-up questions that might interest you: 1. How do you handle quadratic forms when the matrix has complex eigenvalues? 2. What is the relationship between eigenvalues and extremal values in quadratic forms? 3. Could the function $$f(x)$$ be interpreted differently if $$A$$ had real eigenvalues? 4. How can we analyze the extremum of quadratic forms in other contexts, such as optimization? 5. What role does the discriminant play in determining the nature of the eigenvalues? **Tip:** If you encounter quadratic forms with real coefficients but complex eigenvalues, it’s often useful to transform the system into a more suitable form, such as by considering a complex or generalized eigenvalue approach.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Quadratic Forms
Eigenvalues and Eigenvectors
Linear Algebra
Formulas
f(x) = xᵀ A x
Determinant of a matrix
Characteristic equation of a matrix det(A - λI) = 0
Theorems
Eigenvalue-eigenvector theorem
Properties of quadratic forms
Determinant and discriminant analysis
Suitable Grade Level
Grades 11-12
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