Math Problem Statement

Reduce the quadratic form to canonical form by an orthogonal transformation: 3x^2 + 2y^2 + 3z^2 - 2xy - 2yz.

Solution

The quadratic form in the image is:

3x2+2y2+3z2−2xy−2yz3x^2 + 2y^2 + 3z^2 - 2xy - 2yz

To reduce this quadratic form to its canonical form using an orthogonal transformation, we proceed by finding the matrix representation of the quadratic form, diagonalizing it, and using an orthogonal transformation to achieve the canonical form.

1. Construct the matrix:

The quadratic form can be written in matrix form as xTAx\mathbf{x}^T A \mathbf{x}, where AA is a symmetric matrix and x=[xyz]\mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}. By comparing the given quadratic expression with the general form

xTAx=a11x2+a22y2+a33z2+2a12xy+2a13xz+2a23yz,x^T A x = a_{11}x^2 + a_{22}y^2 + a_{33}z^2 + 2a_{12}xy + 2a_{13}xz + 2a_{23}yz,

we identify the matrix AA as:

3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{bmatrix}$$ ### 2. **Diagonalize the matrix**: Next, we need to diagonalize $$A$$. This is done by finding its eigenvalues and eigenvectors, which will allow us to transform the quadratic form into its canonical form. I will compute the eigenvalues and eigenvectors of this matrix, which will then be used to express the quadratic form in terms of new variables obtained through the orthogonal transformation. Let's proceed with the calculation of the eigenvalues and eigenvectors.The eigenvalues of the matrix $$A$$ are: $$\lambda_1 = 1, \quad \lambda_2 = 3, \quad \lambda_3 = 4$$ The corresponding eigenvectors are: $$v_1 = \begin{bmatrix} 0.4082 \\ 0.8165 \\ 0.4082 \end{bmatrix}, \quad v_2 = \begin{bmatrix} -0.7071 \\ 0 \\ 0.7071 \end{bmatrix}, \quad v_3 = \begin{bmatrix} 0.5774 \\ -0.5774 \\ 0.5774 \end{bmatrix}$$ ### 3. **Canonical form**: Using these eigenvalues, the quadratic form in canonical form becomes: $$x'^2 + 3y'^2 + 4z'^2$$ Here, $$x', y', z'$$ are the new variables obtained by applying the orthogonal transformation defined by the eigenvectors. This form is now diagonal and represents the reduced form of the original quadratic expression. Would you like more details on the transformation process or any further steps? ### Related Questions: 1. What is the importance of diagonalizing a quadratic form? 2. How do orthogonal transformations preserve the form of a quadratic equation? 3. How can eigenvalues help in determining the nature of a conic section or surface? 4. What does the canonical form reveal about the geometry of the quadratic form? 5. How can we interpret the physical meaning of diagonalizing a quadratic form? ### Tip: Eigenvectors from the diagonalization process correspond to principal axes, which simplify complex interactions in multidimensional quadratic forms.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Quadratic Forms
Orthogonal Transformations
Diagonalization
Eigenvalues and Eigenvectors

Formulas

Quadratic form in matrix notation: x^T A x
Eigenvalue equation: A v = λ v

Theorems

Spectral Theorem for symmetric matrices
Diagonalization Theorem

Suitable Grade Level

Undergraduate Mathematics

Related Recommendation

Reducing a Quadratic Form to Canonical Form using Orthogonal Transformation

Reduction of Quadratic Form 2x^2 + 2y^2 + 2z^2 - 2xy - 2yz - 2xz to Canonical Form Using Orthogonal Transformation

Reducing Quadratic Forms to Canonical Form

Canonical Form of a Quadratic Form: 2x² + 2y² + 2z² - 2xy + 2xz - 2yz

Orthogonal Reduction of Quadratic Form x1^2 + 5x2^2 + x3^2 + 2x1x2 + 2x2x3 + 6x1x3 to Canonical Form