Math Problem Statement

Consider the quadratic form \( f(x_1, x_2) = 3x_1^2 - 3x_1 x_2 + 2x_2^2 \). (a) Find the symmetric matrix \( A \) such that \( f(x_1, x_2) = [x_1 x_2] A [x_1 x_2]^T \) for all \( x_1, x_2 \). (b) Verify that the eigenvalues and eigenvectors of \( A \) are ... (rest of the question as in the image).

Solution

This question involves analyzing a quadratic form ( f(x_1, x_2) = 3x_1^2

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Quadratic Forms
Eigenvalues and Eigenvectors
Matrix Diagonalization
Positive and Negative Definiteness

Formulas

Quadratic form representation \( f(x) = x^T A x \)
Eigenvalue equation \( A v = \lambda v \)
Diagonalization of a matrix using orthogonal matrices

Theorems

Spectral Theorem for symmetric matrices
Definiteness of a quadratic form

Suitable Grade Level

Undergraduate

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