Exercises – Long Day#
Exercise 1: Eigenvalues and Eigenvectors#
We are given the matrix
Determine whether the following vectors are eigenvectors of matrix \(\mathbf A\). If yes, compute the corresponding eigenvalue.
\(\left[\begin{array}{c} 1 \\ 2\end{array}\right]\)
\(\left[\begin{array}{c} 1 \\ 1\end{array}\right]\)
Hint
One can find the definition of an eigenvector in Definition 11.1.1.
Hint
For each of the two vectors \({\mathbf v}\), investigate whether \({\mathbf A}\cdot {\mathbf v}\) can be written as a scalar multiple of \({\mathbf v}\).
Answer
The vector \(\left[\begin{array}{c} 1 \\ 2\end{array}\right]\) is not an eigenvector of \({\mathbf A}\).
The vector \(\left[\begin{array}{c} 1 \\ 1\end{array}\right]\) is an eigenvector of \({\mathbf A}\). The corresponding eigenvalue is \(5\).
Exercise 2: Real Eigenvalues and Eigenvectors#
This exercise should be solved purely by hand. As in the previous exercise we consider the matrix
Question a#
Determine the characteristic polynomial of \(\mathbf A\,\) and use it to find the eigenvalues of \(\mathbf A\,\). Determine their algebraic multiplicities as well.
Hint
The characteristic polynomial of an \(n \times n\) matrix is defined as \(\mathrm{det}({\mathbf A}-Z\cdot {\mathbf I}_n)\). You can find this in the textbook immediately after Theorem 11.1.1.
Answer
The characteristic polynomial is \(Z^2+Z-30\). The eigenvalues are the roots of this polynomial: \(5\,\) and \(-6\), and hence we have \(Z^2+Z-30=(Z-5)\cdot(Z+6)\). We see that both eigenvalues have an algebraic multiplicity of \(1\).
Question b#
Determine the eigenspace \(E_{-6}\) corresponding to the eigenvalue \(-6\). What is the geometric multiplicity of the eigenvalue \(-6\)?
Hint
In general, if \(\lambda\) is an eigenvalue of an \(n \times n\) matrix \(\mathbf A\), then \(E_\lambda\) equals \(\mathrm{ker}({\mathbf A}-Z\cdot {\mathbf I}_n)\) according to Lemma 11.2.3 in the textbook.
Hint
In order to determine \(E_{-6}\) one can make use of a similar approach as that used in Example 11.2.1 in the textbook. Note that the geometric multiplicity of an eigenvalue is defined in Definition 11.2.1.
Answer
The matrix
has the reduced row-echelon form
Hence it applies that
This shows that the vector \(\left[\begin{array}{c} -10 \\ 1\end{array}\right]\) constitutes a basis for \(E_{-6}\). In particular one can conclude that \(\mathrm{dim}(E_{-6})=1\), which is the same as saying that \(-6\) has a geometric multiplicity of \(1\).
Question c#
The eigenspace corresponding to the eigenvalue \(5\) is denoted \(E_{5}\). Try determining \(\mathrm{dim}(E_5)\) by using Theorem 11.2.4. Use the results from Exercise 1 to provide a basis for \(E_{5}\).
Hint
Theorem 11.2.4 in the textbook implies that \(\mathrm{dim}(E_5)=1\). A basis for \(E_5\) hence contains only one vector.
Answer
The vector \(\left[\begin{array}{c} 1 \\ 1\end{array}\right]\) constitutes a basis for \(E_{5}\).
Exercise 3: Eigenvalues and Eigenvectors in an Infinite-Dimensional Case#
Let \(C_\infty(\Bbb R)\) be the infinite-dimensional, real vector space mentioned in Example 9.3.4 in the textbook. It contains functions from \(\mathbb{R}\) to \(\mathbb{R}\) that can be differentiated an arbitrary number of times. A linear map \(L: C_\infty(\Bbb R) \to C_\infty(\Bbb R)\) is given by \(L(f)=f''\). In other words, \(L\) maps a function \(f \in C_\infty(\Bbb R)\) to its second-order derivative.
Question a#
Determine whether the following functions are eigenvectors of \(L\). If yes, determine the corresponding eigenvalue.
\(f_1(t)=t^2\)
\(f_2(t)=\cos(t)\)
\(f_3(t)=\sin(t)\)
\(f_4(t)=\mathrm e^{4 t}\)
\(f_5(t)=t\mathrm e^t\)
Hint
Investigate for each of the given functions \(f\) whether \(L(f)\) can be written as a scalar multiple of of \(f\).
Answer
The function \(f_1(t)=t^2\) is not an eigenvector of \(L\).
The function \(f_2(t)=\cos(t)\) is an eigenvector of \(L\). The corresponding eigenvalue is \(-1\).
The function \(f_3(t)=\sin(t)\) is an eigenvector of \(L\). The corresponding eigenvalue is \(-1\).
The function \(f_4(t)=\mathrm e^{4t}\) is an eigenvector of \(L\). The corresponding eigenvalue is \(16\).
The function \(f_5(t)=t\mathrm e^t\) is not an eigenvector of \(L\).
Exercise 4: Complex Eigenvalues and Eigenvectors#
We are given the matrix
Question a#
Set up the characteristic polynomial of \(\mathbf A\,\) and use it to compute the eigenvalues of \(\mathbf A\,\).
Hint
The characteristic polynomial of an \(n \times n\) matrix is defined as \(\mathrm{det}({\mathbf A}-Z\cdot {\mathbf I}_n)\). You can also find it in the textbook right after Theorem 11.1.1.
Answer
The characteristic polynomial is \(Z^2-6Z+10\). The eigenvalues are \(\,3+i\,\) and \(\,3-i\,.\)
Question b#
Determine the eigenspace \(E_{3+i}\) corresponding to the eigenvalue \(3+i\). You may use SymPy to solve the linear equation system or for finding the reduced row-echelon form.
Answer
\(E_{3+i}=\mathrm{Span}\left( \left[\begin{array}{cc} 1-i\\ 1\end{array}\right]\right)\,.\)
Question c#
State without further computations the eigenspace that belongs to the other eigenvalue.
Answer
When matrices have real coefficients, one eigenvector/eigenspace is found from the other as the complex conjugate. We thus see that \(E_{3-i}=\mathrm{Span}\left( \left[\begin{array}{cc} 1+i\\ 1\end{array}\right]\right)\,.\)
Question d#
Check your answer with the SymPy command eigenvects() .
Hint
If A is the given \(2 \times 2\) matrix, then the command A.eigenvects() gives a list of two sets as the output. Each set contains the following data: first an eigenvalue, then its algebraic multiplicity, and lastly an ordered basis for the eigenspace.
Exercise 5: A Nice Basis Change#
We consider the same matrix,
as in Exercise 4. Let \(\beta\) be the ordered standard basis for \(\mathbb{C}^2\). We define another ordered basis for \(\mathbb{C}^2\) as follows:
Question a#
Compute the change-of-basis matrix \({}_\beta[\mathrm{id}_{\mathbb{C}^2}]_\gamma\). Now use SymPy to compute the change-of-basis matrix \({}_\gamma[\mathrm{id}_{\mathbb{C}^2}]_\beta\).
Hint
The following equation can be useful: \({}_\gamma [\mathrm{id}_{\mathbb{C}^2}]_\beta = ({}_\beta[\mathrm{id}_{\mathbb{C}^2}]_\gamma)^{-1}.\)
Question b#
Compute (e.g. using SymPy) the matrix \({}_\gamma [\mathrm{id}_{\mathbb{C}^2}]_\beta \cdot {\mathbf B} \cdot {}_\beta [\mathrm{id}_{\mathbb{C}^2}]_\gamma\). The result will be a diagonal matrix. Why, though?
Hint
The columns in \({}_\beta [\mathrm{id}_{\mathbb{C}^2}]_\gamma\) are eigenvectors of matrix \(\mathbf B\).
Question 6: Eigenvalues and Eigenvectors of a Real \(3 \times 3\) matrix#
A linear map \(f: \mathbb{R}^3 \to \mathbb{R}^3\,\) between real vector spaces has with respect to the ordered standard basis for \(\mathbb{R}^3\,\) the following mapping matrix:
Determine the characteristic polynomial and find the eigenvalues of \(f\,\). State the algebraic multiplicity of the eigenvalues. Determine the eigenspaces that belong to each of the eigenvalues, and state the geometric multiplicities of the eigenvalues.
Hint
The characteristic polynomial of a linear map can be determined using Definition 11.1.3 from the textbook. The algebraic and geometric multiplicity is described in Definition 11.2.1.
Hint
The algebraic multiplicity of an eigenvalue \(\,\lambda\,\) is the multiplicity that the root \(\lambda\) has in the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace associated with \(\lambda\) (meaning, the subspace that is spanned by eigenvectors of \(f\) with eigenvalue \(\,\lambda\)).
Hint
Try following the same approach as in Example 11.2.3 in the textbook.
Answer
The eigenvalues are \(2\) with \(\mathrm{gm}(2) = \mathrm{am}(2) = 1\), and \(3\) with \(\mathrm{gm}(3) = \mathrm{am}(3) = 2\). A possible ordered basis for the eigenspace \(E_2\,\) is
A possible ordered basis for the eigenspace \(E_3\,\) is
Exercise 7: Eigenvalues and Eigenvectors of another Real \(3 \times 3\) matrix#
We now consider the matrix
Find the eigenvalues of \(\mathbf B\) and denote their algebraic multiplicities. Determine the real eigenspaces that correspond to each of the eigenvalues, and state the geometric multiplicities of the eigenvalues. You are welcome to use SymPy for computing determinants and reduced row-echelon forms.
Hint
Writing Z=symbol('Z') in SymPy enables the use of Z as a variable. The characteristic polynomial of the matrix \({\mathbf B}\) is computed using this: Type the matrix \({\mathbf B}-Z\cdot {\mathbf I}_3\) into SymPy and then compute its determinant. The determinant of a matrix A can be computed in SymPy as A.det().
Answer
The eigenvalues are \(1\) with \(\mathrm{gm}(1) = 1 < 2 = \mathrm{am}(1)\) and \(-1\) with \(\mathrm{gm}(-1) = \mathrm{am}(-1) = 1\). A possible ordered basis for \(E_{-1}\,\) is
A possible ordered basis for \(E_1\,\) is
Exercise 8: The Characteristic Polynomial of a \(2 \times 2\) Matrix#
Let \(\mathbb{F}\) be a field and \(a,b,c,d\) elements from \(\mathbb{F}\). We consider the matrix
Question a#
Show that the matrix \(\mathbf A\) has the characteristic polynomial \(Z^2-(a+d)Z+\mathrm{det}({\mathbf A})\). Note: The expression \(a+d\) is called the trace of the matrix and is denoted by \(\mathrm{tr}({\mathbf A})\).
Question b#
Assume that \({\mathbf A}\) has eigenvalues \(\lambda_1\) and \(\lambda_2\). Use the previous question to realise that \(\lambda_1+\lambda_2=\mathrm{tr}({\mathbf A})\) and \(\lambda_1 \cdot \lambda_2=\mathrm{det}({\mathbf A})\).
Hint
If a \(2 \times 2\) matrix \({\mathbf A}\) has eigenvalues \(\lambda_1\) and \(\lambda_2\), then it holds true that the characteristic polynomial of the matrix is \((Z-\lambda_1)\cdot (Z-\lambda_2).\)
Answer
On the one hand we have \(p_{\mathbf A}(Z)=Z^2-\mathrm{tr}({\mathbf A})Z+\mathrm{det}({\mathbf A})\). On the other hand we have \(p_{\mathbf A}(Z)=(Z-\lambda_1)\cdot (Z-\lambda_2)=Z^2-(\lambda_1+\lambda_2)Z+\lambda_1\cdot \lambda_2\). Comparing the coefficients will provide what we are seeking.
Exercise 9: Linear Independency of two Eigenvectors#
We are given a matrix \({\mathbf A} \in \mathbb{C}^{2 \times 2}\) as well as two of its eigenvectors \({\mathbf v}_1,{\mathbf v}_2 \in \mathbb{C}^2\). Assume that the corresponding eigenvalues \(\lambda_1,\lambda_2\) are different. Show that \({\mathbf v}_1\) and \({\mathbf v}_2\) are linearly independent.
Hint
If \({\mathbf v}_1\) and \({\mathbf v}_2\) are linearly dependent, then two complex numbers \(c_1\) and \(c_2\), which are not both equal to zero, exist such that
Hint
From the previous hint it follows that if two vectors are linearly dependent, then one of them is a scalar multiple of the other. Is this possible if the two vectors are eigenvectors of the same matrix \({\mathbf A}\) but with different eigenvalues?