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\lhead{{\sf Summer Bridge Course (Algebra)}: Tuesday July 28}
\chead{}\rhead{\thepage}
\lfoot{Due Wednesday, July 29 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}

\begin{document}

%\(\;\) \vspace{-4mm}

\noindent \textbf{Topics}: Eigenvectors and generalized eigenvectors. \\

\noindent \textbf{Do}: Answer the following questions. Assume that all matrices are over the field $K=\mathbb{C}$. \\

\noindent The \emph{characteristic polynomial} of an $n\times n$ matrix $A$ is $p_A(t):=\det(tI-A)$. By the \emph{Cayley Hamilton theorem}, $p_A(A)=0$. The \emph{minimal polynomial} is the smallest-degree monic polynomial $m_A(t)$ for which $m_A(A)=0$, and it must divide $p_A(t)$. \\

%%------------------------------------------------------------------------

%% HW 16, Problem 1
\begin{exercise}
  Consider the following matrices:
  \[
  \A=\begin{bmatrix}7&6\\6&-2\end{bmatrix}\,\qquad\qquad
  \B=\begin{bmatrix}0&-2&2\\-2&-1&0\\2&0&1\end{bmatrix}.
  \]
  \begin{enumerate}[label=(\roman*)]
  \item Find the characteristic and minimal polynomials of each.
  \item Diagonalize each matrix into \(\bold{QDQ}^T\), where
    \(\bold{Q}\) is a (real-valued) \emph{orthogonal} matrix.
  \item Find \emph{all} orthogonal matrices that diagonalize \(\A\). How many will diagonalize \(\B\)? \\
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}
  
%%------------------------------------------------------------------------

%% HW 16, Problem 2
\begin{exercise}
  Do the following for the matrix \(\A\) from the previous worksheet, and then repeat it for \(\B\).
  \[
  \A=\begin{bmatrix}-1&0&1&0\\2&1&2&1\\0&0&-1&0\\4&0&-6&1\end{bmatrix}
  \,,\qquad\quad
  \B=\begin{bmatrix}1&0&0&1\\2&1&0&-4\\1&0&1&-2\\0&0&0&1\end{bmatrix},
  \qquad\quad
  \bold{J}_\lambda=\begin{bmatrix}\lambda&1&&\\&\lambda&\ddots& \\ &&\ddots&1\\ &&&\lambda\end{bmatrix}.
  \] 
  \begin{enumerate}[label=(\roman*)]
  \item For each eigenvalue \(\lambda\), compute \(\dim N_{(\A-\lambda
    \I)^j}\) for \(j=1,2,3,\dots\).
  \item Find the characteristic and minimal polynomials, and all
    (genuine) eigenvectors.
  \item Find a basis \(\mathcal{B}\) of \(\C^4\) consisting of generalized
    eigenvectors, so that the matrix with respect to this basis is
    \(\J=\P^{-1}\A\P\), where \(\J\) is a \emph{Jordan matrix}. This means
    that \(\J\) is block-diagonal formed from \emph{Jordan blocks}
    \(\J_\lambda\); see above. \\
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

 %%------------------------------------------------------------------------ 

%% HW 16, Problem 3
\begin{exercise}
If \(A\colon X\to X\) is a linear map, then a subspace \(Y\subseteq X\) is \(A\)-\emph{invariant} if \(A(Y)\subseteq Y\). Show that for any scalar \(\lambda\in K\), not necessarily an eigenvalue, the subspace \(Y\) is \(A\)-invariant if and only if it is \((A-\lambda I)\)-invariant. 
\end{exercise}

%\begin{solution}
%\end{solution}


 %%------------------------------------------------------------------------ 

\end{document}