91Ó°ÊÓ

A quadratic form is a special type of polynomial expression where the variables are all of degree two. In this context, a quadratic form is usually presented as a function like \( Q(x, y, z) = ax^2 + by^2 + cz^2 + 2pxy + 2qyz + 2rxz \). Notice how it includes terms that are associated with squared variables, like \( x^2, y^2, \) and \( z^2 \), as well as mixed terms (terms that combine two different variables), like \( xy, yz, \) and \( xz \).

The quadratic form given in the exercise is \( 2x^2 - 3y^2 + z^2 - 4xz \). Here we can identify individual coefficients: \( x^2 \) has a coefficient of 2, \( y^2 \) has \(-3\), and \( z^2 \) has 1. The mixed term \( xz \) has a coefficient of \(-4\). These coefficients are crucial as they translate directly into matrix elements. Each term plays a significant role in constructing the symmetric matrix.

When constructing a matrix from a quadratic form, the goal is to represent the quadratic form as a matrix expression: \( \mathbf{x}^T A \mathbf{x} \). Here, \( \mathbf{x} \) is a vector and \( A \) is our symmetric matrix. This symmetric matrix holds the coefficients of the quadratic form.

To develop the matrix \( A \), follow these steps:

• Place the coefficients of \( x^2, y^2, \) and \( z^2 \) on the diagonal of the matrix.
• Divide each coefficient of the mixed terms, like \( xz \), by 2 and place these values symmetrically on the off-diagonal positions.

For the given quadratic form \( 2x^2 - 3y^2 + z^2 - 4xz \), the matrix \( A \) becomes:\[A = \begin{bmatrix} 2 & 0 & -2 \ 0 & -3 & 0 \ -2 & 0 & 1 \end{bmatrix}\] This matrix form is a compact, organized way to present the quadratic form that simplifies many linear algebra operations.

In linear algebra, a symmetric matrix is one that is equal to its transpose. This means that the matrix \( A \) satisfies the condition \( A = A^T \). Symmetry is essential because it ensures that the geometric representation of transformations (often described by these matrices) behaves predictably.

To verify the symmetry of the constructed matrix \( A \), check that it equals its transpose by comparing elements equally spaced across the diagonal:

• Diagonal elements should remain unchanged since they are mirrored across the diagonal.
• Off-diagonal elements must be equal for corresponding positions, so \( A_{ij} = A_{ji} \).

In our case, the matrix: \[A = \begin{bmatrix} 2 & 0 & -2 \ 0 & -3 & 0 \ -2 & 0 & 1 \end{bmatrix}\]demonstrates this symmetry, evident by the fact that both \( A_{13} = -2 \) and \( A_{31} = -2 \), confirming that each symmetric pair of elements is equal. Completing this verification step ensures accurate mathematical modeling and strengthens the understanding of matrix properties.