Math Problem Statement
Solution
The problem asks to analyze the quadratic form:
and determine its sign based on the symmetry of a matrix . The matrix corresponds to the coefficients of this quadratic form.
Step 1: Represent the quadratic form using a symmetric matrix
The quadratic form can be expressed as:
\begin{bmatrix} a & b \\ b & c \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$$ Comparing terms, we find: $$\begin{aligned} a &= 1, \\ b &= -1, \\ c &= 1. \end{aligned}$$ Thus, the symmetric matrix $$A$$ for this quadratic form is: $$A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}.$$ ### Step 2: Check the definiteness of $$A$$ To determine the sign of the quadratic form, we analyze the definiteness of the matrix $$A$$. This involves computing its eigenvalues or examining its leading principal minors. #### (a) Compute the determinant of $$A$$: $$\det(A) = (1)(1) - (-1)(-1) = 1 - 1 = 0.$$ #### (b) Check the trace of $$A$$: $$\text{trace}(A) = 1 + 1 = 2.$$ Since the determinant is $$0$$, $$A$$ is singular, meaning it is not positive definite or negative definite. This suggests the quadratic form is semidefinite. #### (c) Compute eigenvalues of $$A$$: The eigenvalues $$\lambda$$ satisfy: $$\det(A - \lambda I) = 0,$$ $$\det\begin{bmatrix} 1-\lambda & -1 \\ -1 & 1-\lambda \end{bmatrix} = 0.$$ Expanding: $$(1-\lambda)^2 - (-1)(-1) = 0 \implies (1-\lambda)^2 - 1 = 0.$$ Solving: $$(1-\lambda)^2 = 1 \implies 1-\lambda = \pm 1.$$ Thus: $$\lambda = 0 \quad \text{and} \quad \lambda = 2.$$ #### (d) Conclusion about definiteness: Since one eigenvalue is $$0$$ and the other is $$2 > 0$$, the matrix $$A$$ is positive semidefinite. Therefore, the quadratic form $$Q(x)$$ is nonnegative for all $$(x_1, x_2)$$. --- ### Final Answer: The quadratic form $$Q(x) = (x_1)^2 + (x_2)^2 - 2x_1x_2$$ is **nonnegative** (positive semidefinite). --- Would you like further details or have questions about this process? Here are 5 related questions to expand your understanding: 1. What are the conditions for a quadratic form to be positive definite or negative definite? 2. How are eigenvalues related to the definiteness of a matrix? 3. What is the significance of the determinant being zero in this context? 4. How do you verify the signs of eigenvalues geometrically? 5. Can this quadratic form represent a physical system like energy minimization? **Tip:** Symmetric matrices always have real eigenvalues, making them easier to analyze for definiteness.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Quadratic Forms
Linear Algebra
Matrix Eigenvalues
Definiteness of Matrices
Formulas
Quadratic form: Q(x) = x^T * A * x
Matrix determinant: det(A) = a*d - b*c
Eigenvalue equation: det(A - λI) = 0
Theorems
Definiteness of symmetric matrices based on eigenvalues
Properties of quadratic forms
Suitable Grade Level
Undergraduate Mathematics
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