Question 1194394: Given the following quadratic form involving three variables,
Q (x_1, x_2, x_3) = 5(x^2)_1 + 8(x^1)(x^3) + 3(x^2)_2 - 6(x^2)(x^3) + 10(x^2)_3
a. Derive the symmetric matrix associated with Q
b. Determine the definiteness of the matrix you derived in a
Answer by parmen(42)   (Show Source):
You can put this solution on YOUR website! **a. Derive the Symmetric Matrix Associated with Q**
1. **Organize Terms:**
- Rearrange the given quadratic form to group terms involving the same variables:
Q(x₁, x₂, x₃) = 5x₁² + 3x₂² + 10x₃² + 8x₁x₃ - 6x₂x₃
2. **Identify Coefficients:**
- Extract the coefficients of the quadratic terms, linear terms, and cross-product terms:
- Coefficient of x₁²: 5
- Coefficient of x₂²: 3
- Coefficient of x₃²: 10
- Coefficient of x₁x₃: 8 (note: this is the coefficient of both x₁x₃ and x₃x₁)
- Coefficient of x₂x₃: -6 (note: this is the coefficient of both x₂x₃ and x₃x₂)
3. **Construct the Symmetric Matrix:**
- Arrange the coefficients into a symmetric matrix:
A = | 5 0 4 |
| 0 3 -3 |
| 4 -3 10 |
- This matrix is symmetric because the coefficients of the cross-product terms (x₁x₃ and x₂x₃) are placed symmetrically off the main diagonal.
**b. Determine the Definiteness of the Matrix**
To determine the definiteness of the matrix A, we can use the following methods:
1. **Eigenvalue Method:**
- Calculate the eigenvalues of matrix A.
- If all eigenvalues are positive, the matrix is positive definite.
- If all eigenvalues are negative, the matrix is negative definite.
- If some eigenvalues are positive and some are negative, the matrix is indefinite.
- If some eigenvalues are zero and the rest are positive, the matrix is positive semidefinite.
- If some eigenvalues are zero and the rest are negative, the matrix is negative semidefinite.
2. **Leading Principal Minors Method:**
- Calculate the determinants of the leading principal minors of matrix A.
- If all leading principal minors are positive, the matrix is positive definite.
- If the signs of the leading principal minors alternate between positive and negative (starting with negative), the matrix is negative definite.
- If any leading principal minor is zero, the matrix is not definite.
**In this case, it's recommended to use a computational tool (like MATLAB, Python with NumPy/SciPy) to efficiently calculate eigenvalues and determinants for larger matrices.**
**Note:** The definiteness of the matrix A directly relates to the nature of the quadratic form Q(x₁, x₂, x₃). If A is positive definite, Q(x₁, x₂, x₃) will always be positive for any non-zero vector (x₁, x₂, x₃). Similarly, if A is negative definite, Q(x₁, x₂, x₃) will always be negative for any non-zero vector.
I hope this comprehensive explanation helps!
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