Question

Consider the conic \(5 x^{2}-4 x y+2 y^{2}=30\) a. Write the left side in matrix form. b. Diagonalize the coefficient matrix, finding the eigenvalues and eigenvectors. c. Construct the rotation matrix from the information in part b. What is the angle of rotation needed to bring the conic into standard form? d. What is the conic?

Short answer

The given conic’s standard form is an ellipse with the rotation matrix as \[\begin{bmatrix},2/\sqrt{13} & 1\3/\sqrt{13} & 0\end{bmatrix}\]and the angle of rotation as approximately \(56.31^{\circ}\).

Step-by-step solution

Write the Equation in Matrix Form

Firstly, consider the conic equation \(5 x^{2}-4 x y+2 y^{2}=30\). We can write this equation in matrix form as follows:\[\begin{bmatrix}x\y\end{bmatrix}\begin{bmatrix}5 & -2\-2 & 2\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}= 30\]We have our coefficient matrix as \[A = \begin{bmatrix}5 & -2\-2 & 2\end{bmatrix}\]

Diagonalize the Coefficient Matrix

To find the eigenvalues of the matrix A, it is needed to solve the characteristic equation, \(\text{det}(A - \lambda I) = 0\), where I is the identity matrix and \(\lambda\) represent the eigenvalues. We calculate the eigenvalue as follows:\(\text{det}(A - \lambda I) = \text{det}(\begin{bmatrix}5-\lambda & -2\-2 & 2-\lambda\end{bmatrix}) = 0\)This gives the equation \((5 - \lambda)(2 - \lambda) - 4 = 0\), which simplifies to \(\lambda^2 -7\lambda + 10 = 0\). Solving this quadratic equation, we find the roots \(\lambda = 2, 5\), so these are the eigenvalues of matrix A.Next, to find the eigenvectors, we solve the equation \(A v = \lambda v\)For each eigenvalue:For \(\lambda = 2\): We have\(\begin{bmatrix}5-2 & -2\-2 & 2-2\end{bmatrix}v = 0\)which simplifies to \(\begin{bmatrix}3 & -2\-2 & 0\end{bmatrix}v = 0\)Solve this system and one will have \(\mathbf{v}= (2, 3)\) as an eigenvector.Next, for \(\lambda = 5\):This gives\(\begin{bmatrix}5 - 5 & -2\-2 & 2 - 5\end{bmatrix}v = 0\)which simplifies to \(\begin{bmatrix}0 & -2\-2 & -3\end{bmatrix}v = 0\)Solve this system and one will have \(\mathbf{v}= (1, 0)\) as an eigenvector.

Construct the rotation matrix and compute the angle of rotation

The rotation matrix can be constructed from the eigenvectors. Since each column of a rotation matrix are just the normalized eigenvectors of the matrix A, we have:\[R = \begin{bmatrix}2/\sqrt{13} & 1\3/\sqrt{13} & 0\end{bmatrix}\]The angle of rotation \(\theta\) can be calculated from the first column of the rotation matrix. Using the formula \(\theta = \arctan\left(\frac{y}{x}\right)\) (where \(x\) and \(y\) are the first and second elements of the first column in the above rotation matrix, respectively), we get \(\theta = \arctan(\frac{3/\sqrt{13}}{2/\sqrt{13}}) = \arccos(2/\sqrt{13})\) which can be simplified to \(\theta \approx 56.31^{\circ}\)

Identify the Conic

After rotating the conic by \(\theta\), we obtain the standard form of the conic. Since the eigenvalues are both positive and different from each other, the conic is an ellipse.

Key concepts

Matrix Diagonalization

Matrix diagonalization is the process of transforming a given matrix into a diagonal form, where all the elements outside the main diagonal are zero. This is incredibly valuable, as diagonal matrices are much easier to work with, particularly in computations involving powers and exponentials.
Here's a simple breakdown of the process:

• Start with a square matrix, like our coefficient matrix \( A = \begin{bmatrix} 5 & -2 \ -2 & 2 \end{bmatrix} \).
• Find the eigenvalues by solving the characteristic equation \( \text{det}(A - \lambda I) = 0 \), where \( I \) is the identity matrix.
• Solve this equation systematically to find \( \lambda \), which are the eigenvalues.
• With the eigenvalues, proceed to calculate the eigenvectors, which will help us in diagonalizing the matrix.

Once successfully diagonalized, the matrix reveals its intrinsic properties more clearly and can be manipulated or rotated as needed for specific applications, such as conic sections.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are critical concepts in linear algebra, pivotal in various applications like identifying the natural modes of systems or, in our case, rotating conic sections.
Here's what you need to know:

• An **eigenvalue** \( \lambda \) is a scalar that indicates how a transformation changes the magnitude of a vector, the eigenvector.
• An **eigenvector** \( \mathbf{v} \) is a non-zero vector that only changes by a scalar factor when a linear transformation is applied.

For our matrix \( A \):

• The eigenvalues were found to be \( \lambda = 2 \) and \( \lambda = 5 \).
• With the eigenvalue \( \lambda = 2 \), the eigenvector is \( \mathbf{v} = (2, 3) \).
• For \( \lambda = 5 \), the eigenvector is \( \mathbf{v} = (1, 0) \).

By applying these, we not only diagonalize the matrix but also understand how the original conic can be transformed into its standard form.

Rotation Matrix

The rotation matrix allows us to rotate a conic section in 2D space so that it can be easier to analyze, often expressing it in its standard form.
The key steps to construct a rotation matrix are:

• Use the eigenvectors of the matrix \( A \) as the columns of the rotation matrix.
• Normalize these vectors to maintain the orthogonality and proper statistical properties of a rotation matrix.

In our example:

• We constructed the rotation matrix \( R \) as \( \begin{bmatrix} 2/\sqrt{13} & 1 \ 3/\sqrt{13} & 0 \end{bmatrix} \).
• Each eigenvector’s components define the directions for each axis after rotation.
• The rotation angle \( \theta \) can then be determined using the arctangent or arccosine functions, calculated in our case to be approximately \( 56.31^{\circ} \).

Understanding and utilizing the rotation matrix is crucial for adjusting the orientation of conic sections or any geometric transformations.

Standard Form of Conic

Converting a conic section into its standard form simplifies analysis and interpretation. The standard form is more intuitive and indicates the type and properties of the conic.
Here is how you find and interpret the standard form of a conic:

• Once the conic equation is rotated using the rotation matrix, it transforms into its standard form with aligned axes.
• For an equation like ours, with distinct positive eigenvalues, the conic becomes an ellipse.

Recognizing the standard form:

• An ellipse will have the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
• Once identified, properties like axes lengths and orientations are easily derived from the coefficients of the equation.

By transforming the initial equation into standard form, complex geometric problems become much more accessible and understandable.