To determine if the quadratic form is positive definite, we must first express it in matrix form. A quadratic form is typically expressed as:

$Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}$

where $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ and $A$ is the matrix representing the quadratic form. The quadratic form you provided is:

$3x_1^2 + 2x_2^2 - 2\sqrt{2}x_1x_2$

Step 1: Express the quadratic form in matrix form

The general quadratic form is:

$ax_1^2 + bx_2^2 + 2cx_1x_2$

Comparing this with $3x_1^2 + 2x_2^2 - 2\sqrt{2}x_1x_2$, we identify:

• $a = 3$ (coefficient of $x_1^2$)
• $b = 2$ (coefficient of $x_2^2$)
• $c = -\sqrt{2}$ (since the term is $- 2 \cdot \sqrt{2} \cdot x_1x_2$)

The corresponding matrix $A$ for this quadratic form is symmetric and can be written as:

$A = \begin{bmatrix} a & c \\ c & b \end{bmatrix} = \begin{bmatrix} 3 & -\sqrt{2} \\ -\sqrt{2} & 2 \end{bmatrix}$

Step 2: Check if the matrix is positive definite

A matrix is positive definite if all its leading principal minors (determinants of top-left submatrices) are positive:

1. First leading principal minor is just the top-left entry: $A_{11} = 3$ Since $3 > 0$, the first condition is satisfied.

2. Second leading principal minor is the determinant of the full matrix: $\text{det}(A) = \text{det}\begin{bmatrix} 3 & -\sqrt{2} \\ -\sqrt{2} & 2 \end{bmatrix}$ Using the determinant formula for a 2x2 matrix, we compute: $\text{det}(A) = (3)(2) - (-\sqrt{2})(-\sqrt{2}) = 6 - 2 = 4$ Since $4 > 0$, the second condition is also satisfied.

Conclusion:

Since both leading principal minors are positive, the matrix $A = \begin{bmatrix} 3 & -\sqrt{2} \\ -\sqrt{2} & 2 \end{bmatrix}$ is positive definite. Therefore, the quadratic form is positive definite.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What are other criteria to check if a matrix is positive definite?
2. Can a quadratic form be positive definite if the matrix has negative entries?
3. How does the Sylvester’s criterion apply to positive definiteness of larger matrices?
4. What is the significance of positive definiteness in optimization problems?
5. How do we diagonalize a quadratic form?

Tip: A matrix is positive definite if all its eigenvalues are positive, which is another way to check definiteness!