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\homework{8: Jordan Canonical Form}{Due Friday, March 14, by 1:30pm}


Homework problems need to show work and contain proofs in order to receive full credit.  Simply stating an answer is only half of the problem in mathematics; you also need to include an argument that persuades your audience that your answer is correct!  If you have any questions, feel free to contact either Yihan or I via email or office hours.  Have fun!

\begin{enumerate}
\item This problem has you go through an example run of our Jordan canonical form construction. Consider the following matrix:
\begin{align*}
A = \begin{bmatrix} 
14 & -6 & -4 & 0\\
6 & 2 & 0 & -4 \\
-2 & 2 & 4 & 0 \\
-2 & 2 & 0 & 4\\
\end{bmatrix}
\end{align*}
\begin{enumerate}
\item Find a Schur decomposition $A = URU^{-1}$.
\item Now, conjugate $R$ by elementary matrices until it is an upper-triangular matrix $R'$ with the following properties:
\begin{itemize}
\item Whenever $r_{ii} \neq r_{jj}$, $r_{ij} = 0$.
\end{itemize}
\item Conjugate $R'$ by a permutation matrix so that we get an upper-triangular matrix $R''$ whose diagonal is the same as $R$, but in order.
\item Finally, conjugate $R''$ by elementary matrices so that it is a block-diagonal matrix $B$, the blocks of which are all Jordan blocks.   Conclude that $A$ can be written in Jordan normal form.
\end{enumerate}
\item \begin{enumerate}
\item Suppose that $B$ is a $n \times n$ Jordan block of the form 
\begin{align*}
B = \begin{bmatrix}
\lambda & 1 & 0 & \ldots & 0\\
0 & \lambda & 1 & \ldots & 0\\
\vdots & \vdots & \ddots & \ddots & \vdots\\
0 & 0 & \ldots & \lambda & 1 \\
0 & 0 & \ldots  & 0 & \lambda
\end{bmatrix}
\end{align*}
Prove that $B$ is similar to $B^T$.
\item Using 1, prove that if $A$ is a block-diagonal matrix where each block is a Jordan block, then $A$ is similar to $A^T$.
\item Using 2, prove that if $A$ is any matrix, $A$ is similar to $A^T$.
\end{enumerate}
\item  Again, suppose that $B$ is a $n \times n$ Jordan block of the form 
\begin{align*}
B = \begin{bmatrix}
\lambda & 1 & 0 & \ldots & 0\\
0 & \lambda & 1 & \ldots & 0\\
\vdots & \vdots & \ddots & \ddots & \vdots\\
0 & 0 & \ldots & \lambda & 1 \\
0 & 0 & \ldots  & 0 & \lambda
\end{bmatrix}
\end{align*}
\begin{enumerate}
\item Show that $\lambda$ is the only eigenvalue of $B$.  Let $E_\lambda$ denote the collection of all vectors $\vec{v}$ such that $B\vec{v} = \lambda \vec{v}$.  What is the dimension of $E_\lambda$?
\item Show that there is some value of $k$ such that $(B - \lambda I)^k$ is the all-zeroes matrix. Find the smallest value of $k$ for which this is true.
\item Suppose that $C$ is any $n \times n$ matrix, such that $C$ has only one distinct eigenvalue $\lambda$.  Prove that there is some value of $k$ such that $(C - \lambda I)^k$ is the all-zeroes matrix.
\end{enumerate}
\item   Suppose that $B$ is a $4 \times 4$ Jordan block of the form 
\begin{align*}
B = \begin{bmatrix}
\lambda & 1 & 0 & 0\\
0 & \lambda & 1 & 0\\
0 & 0 & \lambda & 1 \\
0 & 0  & 0 & \lambda
\end{bmatrix}_.
\end{align*}
Suppose that $| \lambda | < 1$.  Prove that 
\begin{align*}
\lim_{k \to \infty} B^k = \textrm{ the all-zeroes matrix}.
\end{align*}
\end{enumerate}





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