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\headers{Math343}{Homework 13}{Due Nov 19, 2025}

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\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Show all work; no credit for solutions without work.

\begin{enumerate}

\item {} [5.8.8] Let $A=\mtrix{2}{2 & 1 \\ 4 & 5}$ and let $\vv x_0 = \MTWBO{1}{0}$.  Execute the power method to generate $\vv x_k$ and $\mu_k$ for $k=0,\ldots,4$, keeping 3 decimal places.  What is the estimated eigenvalue/eigenvector pair?

%\item A non-zero vector $\vect{v}$ is a \emph{generalized eigenvector} of $A$ associated with eigenvalue $\lambda$ if $\vect{v}$ is in the null space of $(A-\lambda I)^k$ for some positive integer $k$.  Prove that if $\vect{v}$ is a generalized eigenvector of $A$ associated with $\lambda$ and $A=P^{-1}BP$, then $P\vect{v}$ is a generalized eigenvector of $B$.

\kgmcomment{
Not this semester
\item {}[5.9.\{3,13\}] On any given day, a student is either healthy or ill.  Of the students who are healthy today, 95\% will be healthy tomorrow.  Of the students who are ill today, 55\% will still be ill tomorrow.
\begin{enumerate}
	\item What is the stochastic matrix $P$ that models this situation?
	\item Suppose 20\% of the students are ill on Monday.  What fraction or percentage of the students are likely to be ill on Tuesday?  On Wednesday?
	\item If a student is well today, what is the probability that he or she will be well two days from now?
	\item Find the steady-state vector for $P$.
	\item What is the probability that after many days a specific student is ill?  Does it matter if that person is ill today?
\end{enumerate}

\item {}[5.9.\{15-20\}] True/False.  In the following, $P$ is an $n\times n$ stochastic matrix.  Justify each answer.
\begin{enumerate}
	\item The steady state vector is an eigenvector of $P$.
	\item Every eigenvector of $P$ is a steady state vector.
	\item The all ones vector is an eigenvector of $P^T$.
	\item All stochastic matrices are regular.
\end{enumerate}
}

\item {} Let $\vv u = \mtrix{4}{1 & 2 & 3 & 4}^T$ and $\vv v = \mtrix{4}{1 & -1 & 2 & -2}^T$.  Find a scalar $\alpha$ and a vector $\vv w$ such that $\vv u=\alpha\vv v + \vv w$ and $\vv w$ is orthogonal to $\vv v$.

\item {} [6.1.22] Let $\vv u = \mtrix{3}{u_1 & u_2 & u_3}^T$.  Explain why $\vv u\cdot \vv u\ge 0$ directly from the definition of the dot product.  When is $\vv u\cdot \vv u = 0$?

\item {} [6.1.24] Verify the \emph{parallelogram law} for $\vect{u}$ and $\vect{v}$ in $\RR^n$: $||\vect{u}+\vect{v}||^2 + ||\vect{u}-\vect{v}||^2 = 2||\vect{u}||^2 + 2||\vect{v}||^2$.

%\item {} [6.1.29] Let $W=\Span\{\vect{v}_1,\ldots,\vect{v}_p\}$.  Show that if $\vect{x}$ is orthogonal to each $\vect{v}_j$, then $\vect{x}$ is orthogonal to every vector in $W$.

\item {} [6.1.19] True/False.  Justify your answers.
\begin{enumerate}
	\item $\vv v\cdot \vv v = ||\vv v||^2$
	\item For any scalar $c$, $\vv u\cdot (c\vv v) = c(\vv u\cdot \vv v)$.
	\item If the distance from $\vv u$ to $\vv v$ equals the distance from $\vv u$ to $-\vv v$, then $\vv u$ and $\vv v$ are orthogonal.
	\item For a square matrix $A$, vectors in $\Col A$ are orthogonal to vectors in $\Nul A$.  
	\item If vectors $\vv v_1, \ldots, \vv v_p$ span a subspace $W$ and if $\vv x$ is orthogonal to each $\vv v_j$ for $j\in\{1,\ldots,p\}$, then $\vv x$ is in $W^{\perp}$.
\end{enumerate}

\end{enumerate}
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