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\title{Problem Set 5}
\author{Matrix Theory \& Linear Algebra II}
\date{}
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\begin{document}
\maketitle
\thispagestyle{empty}

\noindent
\begin{enumerate}[(1)]
    \item
    Show that if $p$ is a polynomial with real coefficients and a complex root $\lambda$, then $\bar\lambda$ is also a root.
    Conclude that if $\lambda$ is an eigenvalue of a matrix $T$ with real coefficients, then so is $\bar\lambda$.

    \item
    Find the (real or complex) eigenvalues and eigenvectors of the following matrices. Diagonalize\footnote{To diagonalize $M$ is to find a diagonal matrix $D$ and a matrix $S$ such that $M = SDS^{-1}$} each matrix if possible by finding a basis of eigenvectors.
\[
A = \begin{bmatrix} 2 & 1 \\ -1 & 2 \end{bmatrix}, \quad
B = \begin{bmatrix} -1 & -2 \\ 1 & 1 \end{bmatrix}, \quad
C = \begin{bmatrix} 2 & -1 & 3 \\ 0 & 1 & 1 \\ -1 & 1 & 0 \end{bmatrix}, \quad
D = \begin{bmatrix} 1 & 1 & 1 \\ -1 & 2 & 0 \\ 1 & -1 & 1 \end{bmatrix}
\]

\item 

\begin{itemize}
    \item[(a)] Suppose \( \lambda \in \mathbb{F} \) with \( \lambda \neq 0 \). Prove that \( \lambda \) is an eigenvalue of \( T \) if and only if \( \frac{1}{\lambda} \) is an eigenvalue of \( T^{-1} \).
    \item[(b)] Prove that \( T \) and \( T^{-1} \) have the same eigenvectors.
\end{itemize}

\item 
Given the matrix:


\[
A = \begin{bmatrix} 
2 & 6 & -6 \\ 
0 & 5 & -2 \\ 
0 & 0 & 4 
\end{bmatrix}
\]

\begin{enumerate}[(a)]
\item Find its eigenvalues. Is it possible to find the eigenvalues without computing?

\item Is this matrix diagonalizable? Find out without computing anything.

\item If the matrix is diagonalizable, diagonalize it.
\end{enumerate}

\item 
Let \( V \) be a finite-dimensional vector space over \( \mathbb{C} \) and let \( T \in \mathcal{L}(V) \). Assume that, with respect to some basis \( (v_1, v_2, v_3, v_4) \) of \( V \), we have:


\item 
Show that if an operator $T:V\to V$ is not invertible, then $0$ is an eigenvalue of $T$.

\item
\[
\mathcal{M}(T) = 
\begin{pmatrix}
2+i & 3 & -1 & 4 \\
0 & 1 & 5 & -6 \\
0 & 0 & 2+i & 2 \\
0 & 0 & 0 & -3
\end{pmatrix}
\]


\begin{enumerate}[(a)]
\item Find the eigenvalues of \( T \).
\item Determine if \( T \) is invertible.
\item Explain why, for each \( k \in \{1, 2, 3, 4\} \), we have:


\[
T(\text{span}(v_1, \ldots, v_k)) \subseteq \text{span}(v_1, \ldots, v_k).
\]

\end{enumerate}

\item
\begin{itemize}
    \item[(a)] Give an example of a finite-dimensional complex vector space and an operator \( T \) on that vector space such that \( T^2 \) is diagonalizable but \( T \) is not diagonalizable.
    \item[(b)] Suppose \( \mathbb{F} = \mathbb{C} \), \( k \) is a positive integer, and \( T \in \mathcal{L}(V) \) is invertible. Prove that \( T \) is diagonalizable if and only if \( T^k \) is diagonalizable.
\end{itemize}

\item 
Define \( T \in \mathcal{L}(P_4(\mathbb{R})) \) by \( (Tp)(x) = x p'(x) \) for all \( x \in \mathbb{R} \). Find all eigenvalues and eigenvectors of \( T \).

\item 
Suppose \( T \in \mathcal{L}(V) \) and \( u, w \) are eigenvectors of \( T \) such that \( u + w \) is also an eigenvector of \( T \). Prove that \( u \) and \( w \) are eigenvectors of \( T \) corresponding to the same 

\item Suppose \( T \in \mathcal{L}(V) \) is such that every nonzero vector in \( V \) is an eigenvector of \( T \). Prove that \( T \) is a scalar multiple of the identity
\end{enumerate}

\end{document}