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🔬 Tutorial problems eta

🔬 Tutorial problems eta#

\(\eta\).1#

Consider an \((n \times n)\) Vandermonde matrix [this one can be named :)] of the form

\[\begin{split} V = \begin{bmatrix} 1 & 1 & \cdots & 1 \\ x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\ x_1^{n-1} & x_2^{n-1} & \cdots & x_n^{n-1} \end{bmatrix} \end{split}\]

Show that the determinant of \(V\) is given by

\[ \det(V) = \Pi_{j<i \leqslant n}(x_i-x_j) \]

for the cases \(n=2\), \(n=3\) and \(n=4\)

Properties of the determinants help in finding an elegant solution.

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\(\eta\).2#

For each of the linear maps defined by the following matrices

\[\begin{split} T_1 = \begin{pmatrix} 4/3 & -2/3 & 0 \\ -1/3 & 5/3 & 0 \\ 0 & 0 & -1 \end{pmatrix} \end{split}\]
\[\begin{split} T_2 = \begin{pmatrix} 4 & 0 & 1 \\ -2 & 1 & 0 \\ -2 & 0 & 1 \end{pmatrix} \end{split}\]
\[\begin{split} T_3 = \begin{pmatrix} 5 & 0 & 1 \\ 1 & 1 & 0 \\ -7 & 1 & 0 \end{pmatrix} \end{split}\]
\[\begin{split} T_4 = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & -1 \\ 0 & -1 & 3 \end{pmatrix} \end{split}\]

perform the following tasks:

  1. Find eigenvalues

  2. Find at least one eigenvector for each eigenvalue

  3. Form a new basis from the eigenvectors (normalized or not)

  4. Compute the transformation matrix to the new basis

  5. Find the matrix \(T\) in the new basis and verify that it is diagonal

See example in the lecture notes

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\(\eta\).3#

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

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

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\(\eta\).4#

Consider a quadratic form

\[ Q(x) = x_1^2 + x_2^2 + x_3^2 + 4x_4^2 + 2x_1x_2 - 2x_1x_3 + 4x_1x_4 + 6x_2x_4 - 4x_3x_4 \]

Convert this quadratic form to the canonical form this time by algebraic derivations and not eigenvalue decomposition.

Recall all the involved definitions, in particular that a canonical form of quadratic form is just a linear combination of squares of variables.

The algebraic derivation should aim at completing squares, with the appropriate change of bases to effectively change the variables. For example, an expression \(x^2 + 2xy + y^2\) is a square of a variable \(z = x+y\), which can be attained after the change of bases with the transition matrix

\[\begin{split} T = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \\ T \colon \begin{pmatrix} x \\ y \end{pmatrix} \to \begin{pmatrix} x \\ x+y \end{pmatrix} \end{split}\]

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