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\lhead{{\sf Bridge course (Algebra)}: Monday July 22}
\chead{}\rhead{\thepage}
\lfoot{Due Tuesday, July 23 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
  \pagestyle{fancy}

\begin{document}

%$\;$ \vspace{-4mm}

\noindent \textbf{Topic}: 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 $p_A(t)$.

\begin{enumerate}


%%------------------------------------------------------------------------
\item 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}
      \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}

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

\item 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}
    \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}

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

\item 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{enumerate}

\end{document}
