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\lhead{\textbf{MATH 340 --- Linear Algebra}}
\rhead{\textbf{Homework 5}}
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\begin{center}
  {\LARGE\bfseries Homework 5: Eigenvalues and Diagonalization}\\[10pt]
  \begin{tabular}{rl}
    \textbf{Course:} & MATH 340 --- Linear Algebra \\
    \textbf{Instructor:} & Prof.\ Elena Vasquez \\
    \textbf{Due Date:} & Friday, November 7, 2025, 11:59 PM \\
    \textbf{Student Name:} & \underline{\hspace{5cm}} \\
    \textbf{Student ID:} & \underline{\hspace{5cm}} \\
  \end{tabular}
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{\small\textit{Instructions: Show all work and justify each step. Clearly state any theorems you use. Partial credit will be awarded for correct reasoning even if the final answer is incorrect. Submit via Gradescope by the deadline.}}

\problem[20 points]

Let $A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix}$.

\begin{enumerate}[label=(\alph*)]
  \item Find the characteristic polynomial of $A$.
  \item Find all eigenvalues of $A$.
  \item For each eigenvalue, find a basis for the corresponding eigenspace.
  \item Is $A$ diagonalizable? If so, find an invertible matrix $P$ and a diagonal matrix $D$ such that $A = PDP^{-1}$.
\end{enumerate}

\problem[15 points]

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be the linear transformation defined by
\[
  T(x_1, x_2, x_3) = (2x_1 + x_3,\; x_2,\; x_1 + 2x_3).
\]

\begin{enumerate}[label=(\alph*)]
  \item Find the matrix representation of $T$ with respect to the standard basis.
  \item Find the eigenvalues of $T$.
  \item Determine whether $T$ is diagonalizable and justify your answer.
\end{enumerate}

\problem[15 points]

Prove that if $A$ is an $n \times n$ matrix with $n$ distinct eigenvalues, then $A$ is diagonalizable.

\textit{Hint: Show that eigenvectors corresponding to distinct eigenvalues are linearly independent.}

\problem[15 points]

Let $A$ be a $3 \times 3$ matrix with characteristic polynomial $p(\lambda) = -(\lambda - 2)^2(\lambda + 1)$.

\begin{enumerate}[label=(\alph*)]
  \item What are the eigenvalues of $A$ and their algebraic multiplicities?
  \item What are the possible geometric multiplicities of each eigenvalue?
  \item For each possible case in part (b), determine whether $A$ is diagonalizable and explain why.
  \item Give an explicit example of a $3 \times 3$ matrix with this characteristic polynomial that is \emph{not} diagonalizable.
\end{enumerate}

\problem[15 points]

Consider the matrix
\[
  B = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}.
\]

\begin{enumerate}[label=(\alph*)]
  \item Compute $B^2$ and $B^3$. What do you observe?
  \item Find the eigenvalues of $B$. \textit{(Hint: Use part (a) and the fact that if $\lambda$ is an eigenvalue of $B$, then $\lambda^3$ is an eigenvalue of $B^3$.)}
  \item Compute $B^{100}$.
\end{enumerate}

\problem[20 points]

Let $A$ be a real $n \times n$ symmetric matrix (i.e., $A = A^\top$).

\begin{enumerate}[label=(\alph*)]
  \item Prove that all eigenvalues of $A$ are real.

  \textit{Hint: Consider $A\mathbf{v} = \lambda \mathbf{v}$ and compute $\bar{\mathbf{v}}^\top A \mathbf{v}$ using the fact that $A$ is real and symmetric.}

  \item Prove that eigenvectors corresponding to distinct eigenvalues of $A$ are orthogonal.

  \item Let $A = \begin{pmatrix} 3 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 3 \end{pmatrix}$. Find an orthogonal matrix $Q$ such that $Q^\top A Q$ is diagonal.
\end{enumerate}

\vfill

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  \textit{--- End of Homework 5 ---}
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