🔬 Tutorial problems zeta \zeta

🔬 Tutorial problems zeta \(\zeta\)#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.

\(\zeta\).1#

Derive canonical equations for the three conic sections in Euclidean space by using the properties stated in the definitions for:

  • ellipse

  • parabola

  • hyperbola

For example, for ellipse, assume that the focal points are located at \((c,0)\) and \((-c,0)\), and consider an arbitrary point \((x,y) \in \mathbb{R}^2\) which satisfies the defining property of the ellipse. Then, simplify the expression to arrive at the canonical equation of the ellipse, and give interpretations for the parameters \((a,b)\).

Perform additional task for each curve:

  1. Ellipse: derive expressions for the coordinates of the focal points using standard parameters \(a\) and \(b\)

  2. Parabola: explain the geometric meaning of parameter \(p\)

  3. Hyperbola: verify what change of variables leads to the equation \(xy=1\) for the same curve

Start with the definitions of the conic section, and formulate the defining geometric properties of each curve.

Example of what needs to be done, for a circle. Using the definition that a circle is the set of points equidistant from a fixed point (center), we derive the canonical equation of the circle \(x^2 + y^2 = r^2\).

Let a center be situated in the origin. Then, for an arbitrary point \((x,y) \in \mathbb{R}^2\), the distance from the center is given by the Euclidean norm \(\sqrt{x^2 + y^2}\). The defining property of the circle is that this distance is constant, and equal to the radius \(r\). Therefore, the canonical equation of the circle is

\[ \sqrt{x^2 + y^2} = r \quad \iff \quad x^2 + y^2 = r^2 \]

\(\zeta\).2#

Determine definiteness of the quadratic forms defined with the following matrixes either by Silvester’s criterion or eigenvalue criterion. For the asymmetric matrices use their symmetric part \(\frac{1}{2}(A+A^{T})\) when constructing a quadratic form (see exercise \(\gamma\).2)

\[\begin{split} A_1 = \begin{pmatrix} 5 & 0 & 1 \\ 1 & 1 & 0 \\ -7 & 1 & 0 \end{pmatrix} \end{split}\]
\[\begin{split} A_2 = \begin{pmatrix} 5 & -2 & 3 \\ 0 & 4 & 0 \\ 0 & -1 & 3 \end{pmatrix} \end{split}\]
\[\begin{split} A_3 = \begin{pmatrix} 1 & 0 & 12 \\ 2 & -5 & 0 \\ 1 & 0 & 2 \end{pmatrix} \end{split}\]
\[\begin{split} A_4 = \begin{pmatrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{pmatrix} \end{split}\]
\[\begin{split} A_5 = \begin{pmatrix} -4 & 2 & -6 \\ 2 & -1 & 3 \\ -6 & 3 & -9 \end{pmatrix} \end{split}\]

\(\zeta\).3#

Determine definiteness of the quadratic forms defined with the following matrixes either by Silvester’s criterion or eigenvalue criterion.

\[\begin{split} A_1 = \begin{bmatrix} 4 & 1 & 1 & 1 \\ 1 & 3 & 1 & 1 \\ 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix} \end{split}\]
\[\begin{split} A_2 = \begin{bmatrix} -2 & 0 & 0 & 0 \\ 0 & -3 & 1 & 1 \\ 0 & 1 & -2 & 1 \\ 0 & 1 & 1 & -1 \end{bmatrix} \end{split}\]
\[\begin{split} A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4 \end{bmatrix} \end{split}\]
\[\begin{split} A_4 = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \\ 3 & 4 & 1 & 2 \\ 4 & 3 & 2 & 1 \end{bmatrix} \end{split}\]
\[\begin{split} A_5 = \begin{bmatrix} 1 & -1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \end{split}\]

\(\zeta\).4#

Determine whether each of the following quadratic forms in three variables is positive or negative definite or semidefinite, or indefinite:

  1. \(-x^2 - y^2 - 2z^2 + 2xy\)

  2. \(x^2 - 2xy + xz + 2yz + 2z^2 + 3zx\)

  3. \(-4x^2 - y^2 + 4xz - 2z^2 + 2yz\)

  4. \(-x^2 - y^2 + 2xz + 4yz + 2z^2\)

  5. \(-x^2 + 2xy - 2y^2 + 2xz - 5z^2 + 2yz\)

  6. \(y^2 + xy + 2xz\)

  7. \(-3x^2 + 2xy - y^2 + 4yz - 8z^2\)

  8. \(2x^2 + 2xy + 2y^2 + 4z^2\)

\(\zeta\).5#

Find conditions on \(a\) and \(b\) under which the following matrix

\[\begin{split} \begin{bmatrix} a & 1 & b \\ 1 & -1 & 0 \\ b & 0 & -2 \end{bmatrix} \end{split}\]

is negative definite, negative semidefinite, positive definite, positive semidefinite, and indefinite.

There may be no values of \(a\) and \(b\) for which the matrix satisfies some of these conditions.

Contents