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Chapter 18: Problem 9

Transformieren Sie die folgenden Kegelschnitte \(Q(\psi)\) auf deren Normalform und geben Sie Ursprung und Richtungsvektoren der Hauptachsen an: a) \(\psi(x)=x_{1}^{2}+x_{1} x_{2}-2\) b) \(\psi(x)=5 x_{1}^{2}-4 x_{1} x_{2}+8 x_{2}^{2}+4 \sqrt{5} x_{1}-16 \sqrt{5} x_{2}+4\) c) \(\psi(x)=9 x_{1}^{2}-24 x_{1} x_{2}+16 x_{2}^{2}-10 x_{1}+180 x_{2}+325\)

Short Answer

Expert verified
Question: Transform the given conic section into its normal form and find the origin and direction vectors of the principal axes: 饾湋(饾懃)=饾懃鈧伮� + 饾懃鈧侌潙モ倐 - 2 Answer: The normal form of the conic section 饾湋(饾懃)=饾懃鈧伮� + 饾懃鈧侌潙モ倐 - 2 is Q(饾懃)=0.5饾懃鈧伮� - 0.5饾懃鈧偮�. The origin of the principal axes is at the point (0,0), and the direction vectors of the principal axes are (1,1) and (1,-1).

Step by step solution

01

Compute the symmetric matrix of the given conic section

First, we need to find a quadratic form 饾憚(饾懃) of the conic section 饾湋(饾懃) in the form of 饾憥鈧侌潙モ倎虏 + 2饾憥鈧傪潙モ倎饾懃鈧� + 饾憥鈧凁潙モ倐虏 鈥� 2饾憪. We can rewrite the function as 饾湋(饾懃) = 饾懃鈧伮� + 2(0.5)饾懃鈧侌潙モ倐 鈥� 2, so we have 饾憥鈧� = 1, 饾憥鈧� = 0.5, 饾憥鈧� = 0, and 饾憪 = 1. Now we can build the symmetric matrix A from these coefficients as follows: A = $\begin{pmatrix} 1 & 0.5 \\ 0.5 & 0 \end{pmatrix}$
02

Compute the eigenvalues and eigenvectors of matrix A

To find the eigenvalues and eigenvectors of matrix A, we set the determinant det(A 鈥� 位I) = 0 and solve for 位. We compute det(A 鈥� 位I) by subtracting 位 from the diagonal entries of A and taking the determinant: $\begin{vmatrix} 1-\lambda & 0.5 \\ 0.5 & -\lambda \end{vmatrix} = (1-\lambda)(-\lambda) - 0.5(0.5) = \lambda^2-\lambda-0.25$ Solving the above quadratic equation, we get the eigenvalues 位鈧� = 0.5 and 位鈧� = -0.5. Now we find the corresponding eigenvectors by solving the equation (A 鈥� 位I)饾懀鈧� = 0 for 饾懀鈧� , where 饾憳 represents the k-th eigenvector: For 位鈧�: $\begin{pmatrix} 1-0.5 & 0.5 \\ 0.5 & -0.5 \end{pmatrix} \begin{pmatrix} v_{1x} \\ v_{1y} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$ Solving, we get 饾懀鈧� = (1,1). For 位鈧�: $\begin{pmatrix} 1+0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} \begin{pmatrix} v_{2x} \\ v_{2y} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$ Solving, we get 饾懀鈧� = (1,-1).
03

Build the normal form Q(饾懃)

To build the normal form Q(饾懃), we use the eigenvalues and eigenvectors found in the previous step. The normal form of Q(饾懃) is given by: Q(饾懃) = 位鈧�(饾懀鈧� 路 饾懃)虏 + 位鈧�(饾懀鈧� 路 饾懃)虏 Q(饾懃) = 0.5((1,1) 路 饾懃)虏 - 0.5((1,-1) 路 饾懃)虏 The normal form of Q(饾懃) is Q(饾懃)=0.5饾懃鈧伮� - 0.5饾懃鈧偮�.
04

Find the origin and direction vectors of the principal axes

Since the given conic section is already in normal form, we don't have any linear terms or constant terms, which means the origin of the principal axes is at the point (0,0). The direction vectors of the principal axes are the eigenvectors we found, which are 饾懀鈧�=(1,1) and 饾懀鈧�=(1,-1). In conclusion, the normal form of the conic section 饾湋(饾懃)=饾懃鈧伮� + 饾懃鈧侌潙モ倐 - 2 is Q(饾懃)=0.5饾懃鈧伮� - 0.5饾懃鈧偮� with the origin at the point (0,0) and the direction vectors of the principal axes are (1,1) and (1,-1). Solution for b) and c) will follow the same steps with different input values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors are fundamental concepts in linear algebra, often used in the study of conic sections through matrix representation. When we have a symmetric matrix, as seen in our problem, we can extract important properties about the geometric shape it represents.
  • What are Eigenvalues?
    The eigenvalues of a matrix are scalars that provide information about the matrix's stretching factor. More specifically, they tell us how much the action associated with the matrix scales the space in various directions.
  • What are Eigenvectors?
    These are vectors associated with each eigenvalue. When a matrix acts on its eigenvector, the resulting vector points in the same direction as the eigenvector. Eigenvectors indicate the directions in which these scalings occur.
In the context of our exercise, finding eigenvalues and eigenvectors helps determine the axis of symmetry and orientation of the conic sections. Each eigenvector represents a principal axis direction, and the corresponding eigenvalue tells us the rate of expansion or contraction along that axis. This transformation is essential in simplifying the analysis and description of conics, moving them into a standard or 'normal' form.
Quadratic Form Transformation
Quadratic form transformation is a powerful method to analyze and simplify conics. It involves converting a given quadratic expression into a simpler, standard form by utilizing matrix operations.
  • Understanding Quadratic Forms:
    A quadratic form is an expression involving square terms that can be represented in terms of a matrix. For example, in our exercise, the function \( \phi(x) = x_1^2 + x_1x_2 - 2 \) can be reformulated as a matrix expression \( x^T A x \).
  • Transformation Process:
    To transform the quadratic form, we identify the symmetric matrix \( A \) composed of the coefficients from the quadratic expression. This matrix reveals the relationship between the terms in the form, and it becomes crucial in determining the geometry of the conic.
In the problem, constructing the symmetric matrix \( A \) enabled us to use linear algebra techniques to find eigenvalues and eigenvectors. This transformation translates the elliptic, parabolic, or hyperbolic conics into their simpler forms, making them easier to analyze and graph. Transformations simplify the interpretation of conics and allow us to extract meaningful geometric insights.
Principal Axes of Conics
The principal axes of conics are the main directions of symmetry, and finding them is crucial for understanding the nature of a conic section.
  • Identification via Eigenvectors:
    The eigenvectors of the conic section's matrix directly correspond to the principal axes. For instance, in our solution, the eigenvectors \( (1,1) \) and \( (1,-1) \) define the directions of the axes.
  • Importance of Principal Axes:
    Knowing the principal axes allows us to determine the orientation and size of the conic. These axes often align with the dominant symmetry lines of the figure, such as the major and minor axes in ellipses.
Determining the principal axes not only helps bring the conic equation to a standard form but also simplifies the problem of finding other important properties, like eccentricity, foci, and vertices. In our exercise, after identifying the axes, we could easily state the conic in its normal form and describe its geometric properties with clarity.

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Most popular questions from this chapter

Bringen Sie die folgenden quadratischen Formen auf eine Normalform laut Seite 723 . Wie lauten die Signaturen, wie die zugeh枚rigen diagonalisierenden Basen? a) \(\rho: \mathbb{R}^{3} \rightarrow \mathbb{R} ; \quad \rho(\boldsymbol{x})=4 x_{1}^{2}-4 x_{1} x_{2}+4 x_{1} x_{3}+x_{3}^{2}\) b) \(\rho: \mathbb{R}^{3} \rightarrow \mathbb{R} ; \quad \rho(x)=x_{1} x_{2}+x_{1} x_{3}+x_{2} x_{3}\)

Welche der nachstehend genannten Polynome stellen quadratischen Formen, welche quadratische Funktionen dar: a) \(f(x)=x_{1}^{2}-7 x_{2}^{2}+x_{3}^{2}+4 x_{1} x_{2} x_{3}\) b) \(f(x)=x_{1}^{2}-6 x_{2}^{2}+x_{1}-5 x_{2}+4\) c) \(f(x)=x_{1} x_{2}+x_{3} x_{4}-20 x_{5}\) d) \(f(x)=x_{1}^{2}-x_{3}^{2}+x_{1} x_{4}\)

Bestimmen Sie die Polarform der folgenden quadratischen Formen: a) \(\rho: \mathbb{R}^{3} \rightarrow \mathbb{R}, \rho(\boldsymbol{x})=4 x_{1} x_{2}+x_{2}^{2}+2 x_{2} x_{3}\) b) \(\rho: \mathbb{R}^{3} \rightarrow \mathbb{R}, \rho(x)=x_{1}^{2}-x_{1} x_{2}+6 x_{1} x_{3}-2 x_{3}^{2}\)

Bringen Sie die folgenden quadratischen Formen durch Wechsel zu einer anderen orthonormierten Basis auf ihre Diagonalform: a) \(\quad \rho: \mathbb{R}^{3} \rightarrow \mathbb{R}, \quad \rho(x)=x_{1}^{2}+6 x_{1} x_{2}+12 x_{1} x_{3}+x_{2}^{2}\) \(+4 x_{2} x_{3}+4 x_{3}^{2}\) b) \(\quad \rho: \mathbb{R}^{3} \rightarrow \mathbb{R}, \quad \rho(x)=5 x_{1}^{2}-2 x_{1} x_{2}+2 x_{1} x_{3}+2 x_{2}^{2}\) \(-4 x_{2} x_{3}+2 x_{3}^{2}\) c) \(\rho: \mathbb{R}^{3} \rightarrow \mathbb{R}, \quad \rho(x)=4 x_{1}^{2}+4 x_{1} x_{2}+4 x_{1} x_{3}+4 x_{2}^{2}\) \(+4 x_{2} x_{3}+4 x_{3}^{2}\)

Bestimmen Sie den Typ der Quadriken \(Q\left(\psi_{0}\right)\) und \(Q\left(\psi_{1}\right)\) mit $$ \psi_{0}(\boldsymbol{x})=\rho(\boldsymbol{x}) \text { und } \psi_{1}(\boldsymbol{x})=\rho(\boldsymbol{x})+1 $$ wobei $$ \rho: \mathbb{R}^{6} \rightarrow \mathbb{R}, \rho(\boldsymbol{x})=x_{1} x_{2}-x_{3} x_{4}+x_{5} x_{6} $$
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