To reduce the quadratic form

$3x^2 - 3y^2 - 5z^2 - 2xy - 6yz - 6zx$

into its canonical form using an orthogonal transformation, we need to represent it as a symmetric matrix and then find its eigenvalues and eigenvectors.

Step 1: Represent the Quadratic Form as a Matrix

The given quadratic form can be written in matrix notation as:

$\mathbf{x}^T A \mathbf{x}$

where $\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and $A$ is the symmetric matrix associated with the quadratic form:

3 & -1 & -3 \\ -1 & -3 & -3 \\ -3 & -3 & -5 \end{pmatrix}$$ ### Step 2: Find the Eigenvalues and Eigenvectors of $$A$$ To transform the quadratic form to its canonical form, we need to diagonalize the matrix $$A$$. This involves finding the eigenvalues and corresponding eigenvectors of $$A$$. #### Finding Eigenvalues: The eigenvalues $$\lambda$$ are found by solving the characteristic equation: $$\det(A - \lambda I) = 0$$ where $$I$$ is the identity matrix. #### Finding Eigenvectors: For each eigenvalue, solve the equation: $$(A - \lambda I)\mathbf{v} = 0$$ to find the eigenvectors $$\mathbf{v}$$. ### Step 3: Diagonalize the Matrix The matrix $$A$$ can be diagonalized as: $$A = PDP^{-1}$$ where $$D$$ is a diagonal matrix containing the eigenvalues of $$A$$, and $$P$$ is the matrix whose columns are the normalized eigenvectors of $$A$$. ### Step 4: Write the Canonical Form The quadratic form in the new coordinate system (obtained through the orthogonal transformation) will be: $$\mathbf{y}^T D \mathbf{y}$$ where $$\mathbf{y} = P^{-1}\mathbf{x}$$. ### Performing the Computation: Let's find the eigenvalues and eigenvectors of matrix $$A$$. First, the characteristic polynomial is determined by: $$\det(A - \lambda I) = \det \begin{pmatrix} 3 - \lambda & -1 & -3 \\ -1 & -3 - \lambda & -3 \\ -3 & -3 & -5 - \lambda \end{pmatrix} = 0$$ I'll calculate the eigenvalues and eigenvectors next.### Step 4: Write the Canonical Form The eigenvalues of matrix $$A$$ are: $$\lambda_1 = -8, \quad \lambda_2 = 4, \quad \lambda_3 = -1$$ The corresponding eigenvectors (normalized) are: $$\mathbf{v}_1 = \begin{pmatrix} -0.2673 \\ -0.5345 \\ -0.8018 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} -0.9487 \\ 0.0000 \\ 0.3162 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} 0.1690 \\ -0.8452 \\ 0.5071 \end{pmatrix}$$ These eigenvectors form the columns of the matrix $$P$$, and the diagonal matrix $$D$$ is: $$D = \begin{pmatrix} -8 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & -1 \end{pmatrix}$$ The quadratic form in its canonical form is: $$-8y_1^2 + 4y_2^2 - y_3^2$$ where $$\mathbf{y} = P^{-1}\mathbf{x}$$. ### Summary: The quadratic form $$3x^2 - 3y^2 - 5z^2 - 2xy - 6yz - 6zx$$ can be reduced to the canonical form $$-8y_1^2 + 4y_2^2 - y_3^2$$ using the orthogonal transformation defined by the matrix $$P$$ of eigenvectors. --- Would you like further details on any of these steps or have any questions? Here are five related questions: 1. How do you compute the determinant of a 3x3 matrix? 2. What is the significance of eigenvalues in the context of quadratic forms? 3. How do orthogonal transformations preserve the lengths and angles in vector spaces? 4. What are the physical interpretations of quadratic forms in different fields? 5. How can the diagonalization of matrices be applied in solving systems of differential equations? **Tip:** When working with quadratic forms, diagonalization simplifies complex forms into easily interpretable sums of squares, revealing the intrinsic properties of the system.