The given quadratic form is:

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

To express this in matrix form, we write it as:

$Q(x_1, x_2) = \begin{pmatrix} x_1 & x_2 \end{pmatrix} \begin{pmatrix} a & b \\ b & d \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$

where the matrix

$A = \begin{pmatrix} a & b \\ b & d \end{pmatrix}$

is symmetric, and the quadratic form is given by:

$Q(x_1, x_2) = a x_1^2 + d x_2^2 + 2b x_1 x_2$

Step 1: Identify coefficients

Comparing the original quadratic form $\sqrt{3}x_1^2 + 2x_2^2 - 2\sqrt{2}x_1x_2$ with the general form $a x_1^2 + d x_2^2 + 2b x_1 x_2$, we can identify the coefficients as:

• $a = \sqrt{3}$
• $d = 2$
• $b = -\sqrt{2}$

Thus, the matrix $A$ is:

$A = \begin{pmatrix} \sqrt{3} & -\sqrt{2} \\ -\sqrt{2} & 2 \end{pmatrix}$

Step 2: Check if the matrix is positive definite

A matrix is positive definite if:

1. All its leading principal minors are positive.
2. The eigenvalues are positive.

For a $2 \times 2$ matrix, the leading principal minors are:

• The first minor: $a = \sqrt{3}$
• The determinant of the matrix.

First minor:

The first minor is $\sqrt{3}$, which is positive.

Determinant of $A$:

$\det(A) = (\sqrt{3})(2) - (-\sqrt{2})(-\sqrt{2}) = 2\sqrt{3} - 2 = 2(\sqrt{3} - 1)$

Since $\sqrt{3} \approx 1.732$, we have:

$2(\sqrt{3} - 1) \approx 2(1.732 - 1) = 2(0.732) = 1.464$

The determinant is positive.

Conclusion:

Both the first minor and the determinant are positive, so the matrix is positive definite.

Would you like more details on this solution or any specific part?

Here are some questions you might find interesting:

1. What does it mean geometrically for a matrix to be positive definite?
2. How can we find the eigenvalues of a 2x2 matrix?
3. Can you always determine if a matrix is positive definite by checking its determinant?
4. What does the quadratic form represent in this case?
5. How does the sign of cross-product terms like $x_1x_2$ affect the matrix properties?

Tip: Positive definiteness of a matrix ensures that the associated quadratic form is strictly convex, which has important applications in optimization.