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Exercise 1: Geometric Determination of Eigenvalues and Eigenvectors

Open the GeoGebra sheet $ $ Eigenvalue1

A

We consider the set of plane vectors in an ordinary $\,(O, \mathbf i, \mathbf j)\,$-coordinate system. All vectors are considered to be drawn from the origin. $\,\mathbf F\,$ states the mapping matrix for a linear map $f\,$ with respect to the standard basis. An arbitrary vector $\,\mathbf x\,$ is drawn in blue, while the image vector $\,\mathbf y=f(\mathbf x)\,$ is red.

  1. Right click $\,\mathbf x\,$ and choose Animation on (or on a Mac: Ctrl+klik on $\,\mathbf x\,$ and Animation on). How many times are $\,\mathbf y=f(\mathbf x)\,$ parallel to $\,\mathbf x\,$ during a passage of the circle?

  2. Stop the animation with the undo-button in the tool bar. Move (using the mouse) $\,\mathbf x\,$ to the first position where the two vectors are parallel, and determine the ratio between the length of $\,\mathbf y\,$ and the length of $\,\mathbf x\,$. Use the same procedure on the other positions where the two vectors are parallel.

  3. Explain that one (in general) can determine all eigenvalues for $f$ by letting $\,\mathbf x\,$ pass a semi-circle of (e.g.) radius R$=1\,$.

Open the GeoGebra sheet $ $ Eigenvalue2

B

  1. Rotate $\,\mathbf x\,$ in a semi-circle and read all eigenvalues. Furthermore read for each eigenvalue a corresponding (integer) eigenvector.

  2. Check that the eigenvalues found are roots in the characteristic polynomial (by hand).

  3. Check using paper and pencil that the eigenvectors found are the right ones.

  4. You can change $\,\mF\,$ by moving the column vectors $\,\mathbf s1\,$ and $\,\mathbf s2\,$. Repeat the experiment for the points $1,\,2$ and $3$ above using the following settings of $\,\mF\,$:

$$\begin{matr}{rr}1&0\\\\2&-3\end{matr}\,\,,\, \begin{matr}{rr}3&-1\\\\1&1\end{matr}\,\,\mathrm{and}\,\, \begin{matr}{rr}-2&4\\\\1&-2\end{matr}\,.$$

What are the characteristic differences in each of the three scenarios?

  1. Set $\,\mF\,$ to $\,\begin{matr}{rr}2&2\\-1&4\end{matr}\,.\,$ Rotate $\,\mathbf x\,$ in the semi-circle and read all real eigenvalues.

Exercise 2: Complex Eigenvalues and Eigenvectors

Given the matrix

$$\mA=\begin{matr}{rr}2&2\\\\-1&4\end{matr}\,.$$
C

State the characteristic polynomial for $\mA\,,$ and find using this the eigenvalues for $\mA\,.$

D

State the characteristic matrix for $\mA$ corresponding to one of the eigenvalues, and find using this the eigenspace corresponding to the eigenvalue.

E

State without further computations the eigen-space that corresponds to the other eigenvalue.

F

Check the results using Maple’s Eigenvectors.

Exercise 3: Eigenvalues and Eigenvectors. By Hand

A linear map $\,f: \reel^3\rightarrow\reel^3\,$ is with respect to the ordinary basis in $\,\reel^3\,$ given by the mapping matrix \begin{equation} \mA=\begin{matr}{rrr} 1 & -1 & 1 \\ 2 & 4 & -1 \\ 0 & 0 & 3 \end{matr}\,. \end{equation}

A

Determine the characteristic polynomial and find the eigenvalues for $\,f\,$. State the algebraic multiplicity of the eigenvalues. Determine the real eigen-spaces that correspond to each of the eigenvalues, and state the geometric multiplicity of the eigenvalues.

B

If possible: choose a basis for $\reel^3$ with respect to which the mapping matrix for $f$ becomes a diagonal matrix, and state the diagonal matrix.

Now we consider the matrix \begin{equation} \mB=\begin{matr}{rrr} 1 & 1 & 0 \\ 2 & -1 & -1 \\ 0 & 2 & 1 \end{matr}. \end{equation}

C

Find the eigenvalues for $\mB$ and state their algebraic multiplicity. Determine the real eigenspaces corresponding to each of the eigenvalues, and state the geometric multiplicity of the eigenvalues.

D

If possible: State a regular matrix $\,\mV\,$ and a diagonal matrix $\,\mathbf{\Lambda}\,$ that fulfill

$$\,\mV^{-1}\cdot\mB\cdot\mV=\mathbf{\Lambda}\,.$$

Exercise 4: Linear Stretching in the Plane

Open the GeoGebra sheet $ $ Eigenvalue3.

A

  1. $\,\mF\,$ maps the blue object on the red one. Find by
    moving the column vectors $\,\mathbf s1\,$ and $\,\mathbf s2\,$ a diagonal matrix that maps the red object on wanted, dashed position.

  2. Also consider the maps that correspond to $\,\,\begin{matr}{rr} 3 &0 \\ 0 & -2 \end{matr}\,\,$ and $\,\,\begin{matr}{rr} 1 &0 \\ 0 & 2 \end{matr}\,\,.$

  3. Explain that in general it applies that the diagonal elements in diagonal matrices are eigenvalues for $\,\mathbf F\,$ with $\,\mathbf i\,$ and $\,\mathbf j\,$, respectively, as corresponding eigenvectors. What do the eigenvalues have to do with expansion or contraction in the direction of $\,\mathbf x1\,$ and $\,\mathbf x2\,$, respectively?

Open the GeoGebra sheet Eigenvalue4.

B

  1. Move $\,(\mathbf x1\,$ and $\,\mathbf x2)\,$ such that $\,(\mathbf x1,\mathbf x2)\,$ becomes a new basis consisting of eigenvectors for $f$ , and state the corresponding eigenvalues. Hint: The eigenvectors should be as short as possible when there coordinates are integers.

  2. Which coordinates does the point $\,(6,1)\,$ have in the new $\,(0,\mathbf x1,\mathbf x2)$-coordinate system?

Open the GeoGebra sheet $ $ Eigenvalue5.

C

The blue object is fixed in the $\,(0,\mathbf x1,\mathbf x2)$-coordinate system!

  1. Set the mapping matrix to $\,\,\mF=\begin{matr}{rr} 1 &-2 \\ -1 & 0 \end{matr}\,\,$ by moving the column vectors $\,(\mathbf s1\,$ and $\,\mathbf s2)\,.$

  2. Find by moving $\,(\mathbf x1\,$ and $\,\mathbf x2)\,$ a new basis $\,(\mathbf x1,\mathbf x2)\,$ consisting of eigenvectors for F, and determine the corresponding eigenvalues. State the mapping matrix with respect to the basis $\,(\mathbf x1,\mathbf x2)\,.$ How do you see the relation between the blue and the red object?

  3. Repeat the investigation in the preceding question with mapping matrix that is given in the GeoGebra sheet $ $ Eigenvalue6.

  4. Formulate a consolidated hypothesis about what eigenvalues and their corresponding eigenvectors say about the linear map they stem from.

Exercise 5: Eigenvalues in Functional Spaces

Consider the linear map $\,f:C^{\infty}(\reel)\rightarrow C^{\infty}(\reel)\,$ given by

$$ f(x(t))=x'(t)-x(t)\,.$$
A

Explain that for every $\,k \in \reel\,$ it applies that the function $\,\e^{k\cdot t}\,$ (where $\,t\in \reel\,$) is an eigenvector for $\,f\,,$ and state the corresponding eigenvalue.

B

Explain that the four functions $\,\e^{k\cdot t}\,$ hvor $\,k\in\left{-1,0,1,2\right}\,$ are linearly independent.

Let $\,U\,$ denote the subspace in $\,C^{\infty}(\reel)\,$ that has the basis $\,v=(\e^{-t},\,1,\,\e^t,\,\e^{2\cdot t}\,)\,.$

C

Show that the image space $\,f(U)\,$ is a subspace in $\,U\,,$ and determine the mapping matrix $\,\matind vFv\,$ for the map $f:U\rightarrow U\,$ with respect to basis $\,v\,.$

D

Determine the coordinate vector for

$$\,q(t)=-6\e^{-t}+\e^{2t}+2\,$$

with respect to the basis $\,v\,,$ and find, using the mapping matrix found in the preceding question, all solutions in $\,U\,$ to the equation

$$\,f(x(t))=q(t)\,.$$

E

Compare the result of the preceding question with the outcome of Maple’s dsolve. Why is there not in $\,C^{\infty}(\reel)\,$ more solutions to the equation

$$\,f(x(t))=q(t)\,\,\,\,\mathrm{in} \,C^{\infty}(\reel)\,$$

than there is in the subspace $\,U\,?$

Exercise 6: Diagonalization of a Matrix. Simulated by hand

A linear map $\,f: \reel^3\rightarrow\reel^3\,$ has with respect to the ordinary basis in $\,\reel^3\,$ the mapping matrix \begin{equation} \mA=\begin{matr}{rrr} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 2 \end{matr}. \end{equation}

A

State a basis $\,v\,$ for $\,\reel^3\,$ with respect to which the mapping matrix for $\,f\,$ becomes a diagonal matrix, and state the corresponding basis shift matrix $\,\matind eMv\,$ that changes from $v$-coordinates to $e$-coordinates.

B

State a regular matrix $\,\mV\,$ and a diagonal matrix $\,\mathbf{\Lambda}\,$, such that

$$\mathbf{\Lambda}=\mV^{-1}\cdot\mA\cdot\mV\,.$$

Exercise 7: Diagonalization of a Matrix. Maple

A

Find using Eigenvectors all eigenvalues and the corresponding real eigenspaces for the matrix

$$\,\mB=\begin{matr}{rrrr} -1 & -1 & -6 & 3 \\\\ 1 & -2 & -3 & 0 \\\\ -1 & 1 &0 & 1 \\\\ -1 & -1 & -5 & 3 \end{matr}\,.$$

B

Investigate whether a regular matrix $\,\mV\,$ and a diagonal matrix $\,\mathbf{\Lambda}\,$ exist, such that

$$\mathbf{\Lambda}=\mV^{-1}\cdot\mB\cdot\mV\,.$$

Exercise 8: Eigenvectors Linear Independence. Theory Exercise

Assume that $\,\lambda_1\,$ and $\,\lambda_2\,$ are two different eigenvalues for a matrix $\,\mA\,.$ Then vectors $\,\mv_1\neq 0\,$ and $\,\mv_2\neq 0\,$ exist such that \begin{equation} \mA\cdot\mv_1=\lambda_1\cdot\mv_1\;\mathrm{and}\;\mA\cdot\mv_2=\lambda_2\cdot\mv_2,\quad\mathrm{where}\;\lambda_1\neq\lambda_2\,. \end{equation}

A

Now show that the eigenvectors $\,\mv_1\,$ and $\,\mv_2\,$ are linearly independent.

%####### begin:answer %Argumentationen gennemføres som et modstridsbevis: Antag, at $\mv_1$ og $\mv_2$ er lineært afhængige. Dermed findes der et $k\neq 0$, så $\mv_2=k\cdot\mv_1$. Nu er på den ene side $\mA\cdot\mv_2=\mA\cdot k\cdot\mv_1=k\cdot\lambda_1\cdot\mv_1$ og på den anden side $\mA\cdot\mv_2=\lambda_2\cdot\mv_2=\lambda_2\cdot k\cdot\mv_1$. Ved at sætte højresiderne fra disse to ligninger lig hinanden, opnåes en ny ligning $k\cdot\lambda_1\cdot\mv_1=\lambda_2\cdot k\cdot\mv_1\Rightarrow k(\lambda_1-\lambda_2)\mv_1=\mnul$, hvilket kun kan lade sig gøre, hvis $k=0$ eller $\lambda_1=\lambda_2$, og det strider jo mod de indledende antagelser. Vi kan derfor konkludere, at $\mv_1$ og $\mv_2$ er lineært uafhængige. %####### end:answer

B

Compare the result here obtained with Corollary 13.9 in eNote 13 that is about the linearly independence of eigenvectors.