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\begin{document}

\title{Homework 3
\\\vspace{0.2cm}\scriptsize{Due: Monday, 21 October 2013}
\\\scriptsize{Instructor: Dr. Adam Larios}}
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\noindent \textbf{Solve the problems below, worth 20 points each, for a total of 100 points.  Fully justify your work.}
\begin{enumerate}

\item Derive an expression for the zeros of the minimizing polynomial
\begin{align*}
P_m (x) =\frac{T_m (L(x))}{T_m (L(0))}
\end{align*}

Here $T_m$ is the $m^{\text{th}}$ Chebyshev polynomial and
\begin{align*}
 L(\lambda) = -1 + \frac{2}{\lambda_M-\lambda_0}(\lambda-\lambda_0)
\end{align*}
(Note that there is no matrix involved in this problem, but on our applications, this polynomial was important for a matrix with smallest eigenvalue $\lambda_0$ and large eigenvalue $\lambda_M$.)

\item Let $A\in \nR^{n\times n}$ be a symmetric positive definite.  Show that, for any polynomial $P(\cdot)$, 
\begin{align*}
 \|P(A)\|_A = \rho(P(A))
\end{align*}
for any polynomial $P$.  (Hint: use the definition of the matrix norm and expand everything out in terms of the $(\cdot,\cdot)$-orthonormal eigenvectors of $A$).

\item In many examples, we have seen the $n\times n$ matrix
\begin{align*}
 D = \begin{pmatrix}
      2 & -1 & 0 & 0&\cdots& 0\\
      -1 & 2 & -1 & 0&\cdots &0\\
      \vdots &\vdots &\vdots &\vdots &\ddots&\vdots\\
      0& 0 & 0 &\cdots &2 & -1\\
      0& 0 & 0 &\cdots &-1 & 2\\
     \end{pmatrix}
\end{align*}
Show that $D$ has eigenvalues $\lambda_j=4\sin^2\pnt{\frac{\pi j}{2(n+1)}}$ corresponding to eigenvectors $\psi_j$, whose $k^{th}$ component is given by
\begin{align*}
 (\psi_j)_k = \sin\pnt{\frac{\pi j k}{n+1}}.
\end{align*}
(Hint: Use a basic trig identities.)  
In the case $n=100$, plot the smallest eigenvector and the largest eigenvector (i.e., make the $x$-axis the component number, and the $y$-axis the value of the $k^{th}$ component).  

\item Let $A$ be an SPD matrix and consider the problem $A\bx = \mathbf{b}$.  Suppose instead of minimizing the quadratic functional $q(\bx) = (A\bx,\bx)-2(\mathbf{b},\bx)$ to solve this problem, we instead tried to minimize 
\begin{align*}
 J(\bx) =\frac{1}{2}\|A\bx - \mathbf{b}\|_{L^2}^2
\end{align*}
where $\|\cdot\|_{L^2}$ is the standard Euclidean norm.  What can you say about such an approach?  Will this yield a useful method  for solving $A\bx = \mathbf{b}$?

\item Given a linearly-independent set of vectors $\set{\br_0,\br_1,\ldots,\br_k}$ and an SPD matrix $A$, show that one can construct an $A$-orthonormal set of vectors  $\set{\bp_0,\bp_1,\ldots,\bp_k}$ such that 
\begin{align*}
 \text{span}(\set{\br_0,\br_1,\ldots,\br_k}) = \text{span}(\set{\bp_0,\bp_1,\ldots,\bp_k}).
\end{align*}
Recall that being $A$-orthonormal means that $(A\bp_i,\bp_j) = \delta_{ij}$.  (Hint: Start by choosing $\bp_0$ as a multiple of $\br_0$.  Then assume all $\bp_m$ have been chosen, and show how to construct $\bp_{m+1}$.)


\end{enumerate}

  

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