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\fancyhead[L]{\textbf{MATH 510 Homework 8\qquad\qquad Name:}}
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{\small
\begin{itemize}
\item Due April 12 at the beginning of class.
\end{itemize}}

\begin{enumerate}[{\rm 1.}]
\item If $A\in M_n$ is Hermitian, show that the following three optimization problems all have the same solution:
\begin{enumerate}[{\rm (a)}]
\item $\displaystyle\max_{x^\ast x=1} x^\ast A x$
\item $\displaystyle\max_{x^\ast x\neq 0} \frac{x^\ast A x}{x^\ast x}$
\item $\displaystyle\max_{x^\ast A x=1} \frac{1}{x^\ast x}$ when at least one eigenvalue of $A$ is positive
\end{enumerate}
\item Let $A\in M_n$ have eigenvalues $\{\la_i\}$. Show that, even if $A$ is not Hermitian, one has the bounds
\[\min_{x\neq 0} \left|\frac{x^\ast Ax}{x^\ast x}\right|\le |la_i|\le\max_{x\neq 0}\left|\frac{x^\ast Ax}{x^\ast x}\right|\]
\item Use Weyl's Theorem to show that for Hermitian matrices $A,B\in M_n$ one has
\[|\la_k(A+B)-\la_k(A)|\le\rho(B),\qquad \forall k=1,2,\ldots,n.\]
(This is an example of a \emph{pertubation theorem} for the eigenvalues of a Hermitian matrix.)
\item 
A \emph{hyperellipsoid} in $\mathbb{R}^m$ with radii $a_1,\dotsc,a_m > 0$ is an image of the set
\[
E := \Big\{ (z_1,\dotsc,z_m) \in \mathbb{R}^m : \frac{z_1^2}{a_1^2} + \cdots + \frac{z_m^2}{a_m^2} = 1 \Big\}
\]
under an orthogonal transformation.
A \emph{solid hyperellipsoid} with radii $a_1,\dotsc,a_m > 0$ is an image of the set
\[
B := \Big\{ (z_1,\dotsc,z_m) \in \mathbb{R}^m : \frac{z_1^2}{a_1^2} + \cdots + \tfrac{z_m^2}{a_m^2} \leq 1 \Big\}
\]
under an orthogonal transformation.
Let $T:\mathbb{R}^n \to \mathbb{R}^m$ be a \emph{surjective} linear transformation.
Prove that the image of the unit sphere
\[
S := \{ x \in \mathbb{R}^n : \| x \| = 1 \} \subseteq \mathbb{R}^n
\]
under $T$ is either a hyperellipsoid (if $T$ is injective) or a solid hyperellipsoid (if $T$ is not injective).


\item Let $A\in M_n$ be Hermitian. Let $\la_1\le\la_2\le\cdots\le\la_n$ be the eigenvalues of $A$, and $\la_{i,1}\le\la_{i,2}\le\cdots\le\la_{i,n-1}$ the eigenvalues of the $(n-1)\times(n-1)$ principal submatrix $A{i}'$ obtained from $A$ by deleting the $i$:th column and $i$:th row. Show that the following interlacing inequalities hold:
\[
\begin{tikzpicture}
\node at (0,0) {$\la_1$};
\node[rotate=-45] at (.5,-.5) {$\le$};
\node at (1,-1) {$\la_{i,1}$};
\node[rotate=45] at (1.5,-.5) {$\le$};
\node at (2,0) {$\la_2$};
\node[rotate=-45] at (2.5,-.5) {$\le$};
\node at (3,-1) {$\la_{i,2}$};
\node[rotate=45] at (3.5,-.5) {$\le$};
\node at (4,0) {$\cdots$};
\node[rotate=-45] at (4.5,-.5) {$\le$};
\node at (5,-1) {$\la_{i,n-1}$};
\node[rotate=45] at (5.5,-.5) {$\le$};
\node at (6,0) {$\la_n$};
\end{tikzpicture}
\]
\end{enumerate}
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