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A quadratic form is a polynomial with terms all of degree two. In our case, the quadratic form is given by the equation: \(7 x_1^2 - 24 x_1 x_2 = 144\). This equation is used to represent different types of conic sections, such as ellipses, hyperbolas, and parabolas, which are significant in geometry and many practical applications.
In general, a quadratic form can be expressed in matrix form as \( \mathbf{x}^{\top} A \mathbf{x} = c \), where \( A \) is a symmetric matrix and \( c \) is a constant. For this exercise, the matrix is \( A = \begin{bmatrix} 7 & -12 \ -12 & 0 \end{bmatrix}\) and the constant \( c \) is 144.
When dealing with conic sections, it's important to identify the type of conic represented by the quadratic form. We do this by analyzing the eigenvalues of matrix \( A \). The signs and values of the eigenvalues indicate the shape and orientation of the conic section.

Eigenvalues and eigenvectors provide crucial insights into the structure and characteristics of a matrix, which are used for simplifying the quadratic form into its principal axes. The eigenvalues of matrix \( A \) are determined by finding the roots of its characteristic polynomial: \( |A - \lambda I| = 0\).
For the matrix \( A = \begin{bmatrix} 7 & -12 \ -12 & 0 \end{bmatrix} \), calculate the characteristic polynomial, \( \lambda^2 - 7\lambda - 144 = 0\), which results in the eigenvalues \( \lambda_1 = 18 \) and \( \lambda_2 = -11\). These eigenvalues help classify the conic: one positive and one negative indicates a hyperbola.
Eigenvectors are used to construct the transformation matrix. They are solutions to the equation \( (A - \lambda I) \mathbf{v} = \mathbf{0}\). For \( \lambda_1 = 18 \), we have eigenvector \( \mathbf{v}_1 = \begin{bmatrix} 2 \ 3 \end{bmatrix}\), and for \( \lambda_2 = -11 \), eigenvector \( \mathbf{v}_2 = \begin{bmatrix} -12 \ 7 \end{bmatrix}\). These eigenvectors represent the directions of the axes in the principal axes transformation.

The transformation of axes is a process to simplify conics by rotating and translating them until they align with the coordinate axes. This reveals the nature of the conic more clearly. To perform this transformation, we use the eigenvectors to construct the transformation matrix \( P \). For our matrix \( A = \begin{bmatrix} 7 & -12 \ -12 & 0 \end{bmatrix} \), the transformation matrix is:\

• \( P = \begin{bmatrix} 2 & -12 \ 3 & 7 \end{bmatrix} \)

With \( P \), we rotate the coordinate system so that the new system aligns with the eigenvectors. This process, also known as a change of basis, simplifies our analysis by transforming the quadratic form into a more manageable diagonal form. The transformed equation, aligned with new coordinates \( \mathbf{y} \), is much easier to analyze and graph because the cross-product terms like \( x_1 x_2 \) are eliminated.
Mathematically, this transformation is denoted as \( \mathbf{x} = P \mathbf{y} \). Thus, any point \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) in original coordinates can be expressed in terms of new coordinates \( \mathbf{y} = \begin{bmatrix} y_1 \ y_2 \end{bmatrix} \).

Diagonalization is the process of converting a matrix into a diagonal matrix using a similarity transformation. This is particularly useful in simplifying quadratic equations to their principal axis forms, leading to more straightforward interpretation and solutions.
The diagonal matrix \( D \) results from the transformation \( P^{-1}AP = D \), where \( P \) is the matrix whose columns are the eigenvectors of \( A \). For matrix \( A \), the diagonal matrix is \( D = \begin{bmatrix} 18 & 0 \ 0 & -11 \end{bmatrix} \).
Diagonal matrices are important because:

• They make it easier to identify the type of conic section represented by the quadratic form by examining the signs of the diagonal elements.
• The diagonal elements of \( D \) correspond to the eigenvalues, illustrating the inherent properties of the original matrix.

By diagonalizing the matrix \( A \), we achieve a new simplified quadratic equation in terms of \( \mathbf{y} \), which is easier to analyze and graph. The absence of off-diagonal terms simplifies the discussion of the nature and orientation of the conic.