https://file.aiquickdraw.com/mathbot/file/1728739422970w4mm60sx.jpg

The given expression is:

\[
x^2 - y^2 + 4z^2 + 4xy + 2yz + 6xz
\]

We are tasked with proving that this expression is **indefinite**. To determine if a quadratic form is indefinite, we examine its corresponding matrix and check its signature, which will tell us if the form can take both positive and negative values.

### Step 1: Express the quadratic form in matrix notation

The general quadratic form in three variables \( x \), \( y \), and \( z \) can be written as:

\[
Q(x, y, z) = [x, y, z] \begin{bmatrix} a & b & c \\ b & d & e \\ c & e & f \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}
\]

We want to identify the coefficients \( a \), \( b \), \( c \), \( d \), \( e \), and \( f \) from the given quadratic form:

\[
x^2 - y^2 + 4z^2 + 4xy + 2yz + 6xz
\]

- \( x^2 \) has a coefficient of \( 1 \), so \( a = 1 \).
- \( y^2 \) has a coefficient of \( -1 \), so \( d = -1 \).
- \( z^2 \) has a coefficient of \( 4 \), so \( f = 4 \).
- \( xy \) has a coefficient of \( 4 \), so \( b = 2 \) (since it appears as \( 4xy \), the factor is shared by both).
- \( yz \) has a coefficient of \( 2 \), so \( e = 1 \).
- \( xz \) has a coefficient of \( 6 \), so \( c = 3 \).

Thus, the corresponding matrix for the quadratic form is:

\[
A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & -1 & 1 \\ 3 & 1 & 4 \end{bmatrix}
\]

### Step 2: Compute the determinant and minors

We calculate the principal minors to check if the matrix is indefinite.

1. **First minor** (top left element): 
\[
\Delta_1 = 1
\]

2. **Second minor** (determinant of top-left \( 2 \times 2 \) submatrix):
\[
\Delta_2 = \det \begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix} = (1)(-1) - (2)(2) = -1 - 4 = -5
\]

3. **Third minor** (determinant of the full matrix):
\[
\Delta_3 = \det \begin{bmatrix} 1 & 2 & 3 \\ 2 & -1 & 1 \\ 3 & 1 & 4 \end{bmatrix}
\]

Using cofactor expansion along the first row:

\[
\Delta_3 = 1 \times \det \begin{bmatrix} -1 & 1 \\ 1 & 4 \end{bmatrix} - 2 \times \det \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix} + 3 \times \det \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}
\]

First term:
\[
\det \begin{bmatrix} -1 & 1 \\ 1 & 4 \end{bmatrix} = (-1)(4) - (1)(1) = -4 - 1 = -5
\]

Second term:
\[
\det \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix} = (2)(4) - (1)(3) = 8 - 3 = 5
\]

Third term:
\[
\det \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix} = (2)(1) - (-1)(3) = 2 + 3 = 5
\]

Now substitute back:

\[
\Delta_3 = 1 \times (-5) - 2 \times 5 + 3 \times 5 = -5 - 10 + 15 = 0
\]

### Step 3: Conclusion

The matrix has minors of mixed signs, indicating that the quadratic form is **indefinite**. Specifically, the form can take both positive and negative values for different input vectors, satisfying the condition of indefiniteness.

Would you like further details or have any other questions?

---

Here are 5 related questions to deepen your understanding:

1. What is the significance of the determinant in a quadratic form?
2. How can we classify a quadratic form as positive definite, negative definite, or indefinite?
3. What role do eigenvalues play in determining the definiteness of a quadratic form?
4. How can we geometrically interpret a quadratic form being indefinite?
5. How do the signs of the principal minors help in classifying the definiteness?

**Tip**: When working with quadratic forms, always try to express them in matrix form to analyze their properties effectively through determinants and eigenvalues.

Prove that x^2 - y^2 + 4z^2 + 4xy + 2yz + 6xz is indefinite.

Learn how to prove that the quadratic form x^2 - y^2 + 4z^2 + 4xy + 2yz + 6xz is indefinite. This solution involves analyzing the matrix representation of the quadratic form using Sylvester's criterion.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation