Question

Find the equations of the following conics and quadric surfaces relative to principal axes. $$5 x^{2}+3 y^{2}+2 z^{2}+4 x z=14$$

Short answer

The principal axes coordinates for the given ellipse equation are defined by transforming the original coordinates using the eigenvectors of the coefficient matrix.

Step-by-step solution

- Identify the general form of the quadric surface equation

The given equation is: \[ 5x^{2} + 3y^{2} + 2z^{2} + 4xz = 14 \]Recognize that this is a general second-degree polynomial for quadric surfaces.

- Rewrite the equation in matrix form

We can write the quadratic form as: \[ \begin{pmatrix} x & y & z \ \textbf{} \textbf{} \textbf{} \textbf{}\textbf{}\textbf{} \end{pmatrix} \begin{pmatrix} 5 & 0 & 2 \ 0 & 3 & 0 \ 2 & 0 & 2 \ \textbf{} \textbf{} \end{pmatrix} \begin{pmatrix} x \ y \ z \ \textbf{} \textbf{} \textbf{} \end{pmatrix} = 14 \]

- Determine the coefficient matrix

The coefficient matrix (A) from our quadratic terms, including off-diagonal terms, is:\[ A = \begin{pmatrix} 5 & 0 & 2 \ 0 & 3 & 0 \ 2 & 0 & 2 \ \textbf{} \end{pmatrix} \]

- Find the eigenvalues and eigenvectors

Solve for the eigenvalues and eigenvectors: Compute the characteristic polynomial from \( \text{det}(A - \lambda I) \). Solve for \( \lambda \) by finding the roots of the determinant.

- Diagonalization of the coefficient matrix

Using the eigenvectors, form the diagonal matrix \( D \) and the transformation matrix \( P \). \[ D = P^{-1}AP \]

- Write the transformed equation

Rewrite the original equation using the principal axes: \[ \lambda_{1} x'^{2} + \lambda_{2} y'^{2} + \lambda_{3} z'^{2} = 14 \]

Key concepts

Eigenvalues and Eigenvectors

In this problem, finding the eigenvalues and eigenvectors of the coefficient matrix is crucial to understanding the quadric surface. An eigenvalue is a scalar that changes the magnitude but not the direction of an eigenvector when the matrix transforms that eigenvector.
To find the eigenvalues, solve the characteristic polynomial equation \(\text{det}(A - \text{{\textbackslash}}lambda I) = 0\) where \(A\) is the coefficient matrix and \(\text{{\textbackslash}}lambda\) represents the eigenvalues.
Next, compute the eigenvectors, which are the non-zero vectors \(v\) that satisfy \(Av = \text{{\textbackslash}}lambda v\). These vectors provide the directions in which the quadratic form can be simplified. This simplification is indispensable for understanding the geometric properties and orientation of the quadric surface.
The eigenvalues give you insights about the stretching or compressing of the surface along specific directions.
Eigenvectors, on the other hand, define the new coordinate axes for the principal axes transformation. This makes the quadric surface equation much easier to analyze.

Matrix Diagonalization

Matrix diagonalization is key to converting a complex quadratic form into a simpler one. Once you have the eigenvalues and eigenvectors, you can diagonalize the coefficient matrix. This process involves forming a diagonal matrix \(D\), which contains the eigenvalues on its diagonal entries, and a transformation matrix \(P\), which is formed from the eigenvectors.
The relationship between these matrices is given by \(D = P^{ -1}AP\). Transforming the original coefficient matrix \(A\) into a diagonal matrix \(D\) simplifies the quadratic form.
In the diagonal form, the quadratic surface equation can be written using the principal axes. Here, \(P^{ -1}\) represents the inverse of the transformation matrix. Matrix diagonalization is extremely helpful in simplifying the equations of quadric surfaces, allowing us to express them relative to their principal axes with ease.

Principal Axes Transformation

Principal axes transformation is the step where we use the results from eigenvalues and matrix diagonalization to rewrite the equation of a quadric surface in its simplest form. This transformation aligns the coordinate system with the principal axes of the surface, which are the directions indicated by the eigenvectors of the coefficient matrix.
When the quadratic equation is rewritten using the principal axes, its form becomes significantly simpler. In this case, you align the system of coordinates so that the matrix \(A\) is diagonal, which means there are no cross-product terms like \(xz\).
The transformed equation takes the form: \( \text{{\textbackslash}}lambda_{1} x'^{2} + \text{{\textbackslash}}lambda_{2} y'^{2} + \text{{\textbackslash}}lambda_{3} z'^{2} = 14 \) where \( \text{{\textbackslash}}lambda_{1}, \text{{\textbackslash}}lambda_{2}, \text{{\textbackslash}}lambda_{3} \) are the eigenvalues, and \( x', y', z' \) are the new coordinates along the principal axes.
This makes it far more straightforward to interpret the geometry of the surface, identify its type (ellipsoid, hyperboloid, etc.) and analyze its properties.