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Mathematics LibreTexts

12.4: Review Problems

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    2080

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    1. Try to find more solutions to the vibrating string problem \(\partial^{2} y/\partial t^{2}=\partial^{2} y/\partial x^{2}\) using the ansatz

    \[
    y(x,t)=\sin(\omega t) f(x)\, .
    $$
    What equation must \(f(x)\) obey? Can you write this as an eigenvector equation? Suppose that the string has length \(L\) and \(f(0)=f(L)=0\). Can you find any solutions for \(f(x)\)?

    2. Let \(M=\begin{pmatrix}
    2 & 1 \\
    0 & 2
    \end{pmatrix}\). Find all eigenvalues of \(M\). Does \(M\) have two linearly independent} eigenvectors? Is there a basis in which the matrix of \(M\) is diagonal? \(\textit{i.e.}\), can \(M\) be diagonalized?)

    3. Consider \(L \colon \Re^{2}\rightarrow \Re^{2}\) with $$L\begin{pmatrix}x\\y\end{pmatrix}
    =\begin{pmatrix}x\cos \theta +y\sin \theta\\ -x\sin \theta + y\cos \theta\end{pmatrix}\, .$$
    a) Write the matrix of \(L\) in the basis \(\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\).

    b) When \(\theta\neq 0\), explain how \(L\) acts on the plane. Draw a picture.

    c) Do you expect \(L\) to have invariant directions?

    d) Try to find real eigenvalues for \(L\) by solving the equation

    \[L(v)=\lambda v.\]

    e) Are there complex eigenvalues for \(L\), assuming that \(i=\sqrt{-1}\) exists?

    4. Let \(L\) be the linear transformation \(L \colon \Re^{3}\rightarrow \Re^{3}\) given by
    $$L\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix} x+y \\ x+z \\ y+z\end{pmatrix}.$$
    Let \(e_{i}\) be the vector with a one in the \(i\)th position and zeros in all other positions.

    a) Find \(Le_{i}\) for each \(i\).

    b) Given a matrix \(M=\begin{pmatrix}
    m^{1}_{1} & m^{1}_{2} & m^{1}_{3}\\
    m^{2}_{1} & m^{2}_{2} & m^{2}_{3}\\
    m^{3}_{1} & m^{3}_{2} & m^{3}_{3}\\
    \end{pmatrix}\), what can you say about \(Me_{i}\) for each \(i\)?

    c) Find a \(3\times 3\) matrix \(M\) representing \(L\). Choose three nonzero vectors pointing in different directions and show that \(Mv=Lv\) for each of your choices.

    d) Find the eigenvectors and eigenvalues of \(M.\)

    5. Let \(A\) be a matrix with eigenvector \(v\) with eigenvalue \(\lambda\). Show that \(v\) is also an eigenvector for \(A^{2}\) and what is its eigenvalue? How about for \(A^{n}\) where \(n \in \mathbb{N}\)? Suppose that \(A\) is invertible. Show that \(v\) is also an eigenvector for \(A^{-1}\).

    6. A \(\textit{projection}\) is a linear operator \(P\) such that \(P^{2} = P\). Let \(v\) be an eigenvector with eigenvalue \(\lambda\) for a projection \(P\), what are all possible values of \(\lambda\)? Show that every projection \(P\) has at least one eigenvector.

    Note that every complex matrix has at least 1 eigenvector, but you need to prove the above for \(\textit{any}\) field.

    7. Explain why the characteristic polynomial of an \(n\times n\) matrix has degree \(n\). Make your explanation easy to read by starting with some simple examples, and then use properties of the determinant to give a \(\textit{general}\) explanation.

    8. Compute the characteristic polynomial \(P_{M}(\lambda)\) of the matrix $$M=\begin{pmatrix}
    a & b \\
    c & d
    \end{pmatrix}\, .$$ Now, since we can evaluate polynomials on square matrices, we can plug \(M\) into its characteristic polynomial and find the \(\textit{matrix} P_{M}(M)\). What do you find from this computation?
    Does something similar hold for \(3 \times 3\) matrices? (Try assuming that the matrix of $M$ is diagonal to answer this.)

    9. \(\textit{Discrete dynamical system.}\) Let \(M\) be the matrix given by \[M=
    \begin{pmatrix}
    3 & 2 \\
    2 & 3 \\
    \end{pmatrix}.\]
    Given any vector \(v(0)=
    \begin{pmatrix}
    x(0) \\
    y(0) \\
    \end{pmatrix},\) we can create an infinite sequence of vectors \(v(1), v(2), v(3),\) and so on using the rule:
    \[v(t+1)=M v(t) \text{ for all natural numbers } t.\] (This is known as a {\it discrete dynamical system} whose {\it initial condition} is \(v(0).\))

    a) Find all eigenvectors and eigenvalues of \(M.\)

    b) Find all vectors \(v(0)\) such that \[v(0)=v(1)=v(2)=v(3)=\cdots\] (Such a vector is known as a {\it fixed point} of the dynamical system.)

    c) Find all vectors \(v(0)\) such that \(v(0), v(1), v(2), v(3), \ldots\) all point in the same direction. (Any such vector describes an {\it invariant curve} of the dynamical system.)

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