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\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due April 17 (Friday) }}
\rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear Algebra $|$ Spring
    2025 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-2mm}

\begin{enumerate}
  
%%------------------------------------------------------------------------
  
\item Let $N\colon X\to X$ be a normal mapping of an inner product
  space.  Prove that $||N||=\max|n_i|$, where the $n_i$s are the
  eigenvalues of $N$. \medskip

%%------------------------------------------------------------------------
  
\item Prove that a normal mapping $N\colon X\to X$S has a square-root,
  that is, a matrix $R$ such that $N=R^2$. Is $R$ necessarily normal?
  Unique? \medskip

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

  
  
\item Let $H\colon X\to X$ be self-adjoint, with eigenvalues
  $\lambda_1\geq\cdots\geq\lambda_n$. Prove the following
  \emph{max-min} principle:
  \[
  \lambda_k=\max_{\dim S=k}\min_{x\in S\setminus 0} R_H(x).
  \]

%%------------------------------------------------------------------------
  
\item For any positive mapping $M\colon X\to X$, define an inner
  product on $X$ by $\<x,y\>:=(x,My)$. Throughout this problem, assume
  that $X=\R^2$ and $M=\begin{bmatrix}2 & -1 \\ -1 & 2\end{bmatrix}$.
  \begin{enumerate}
  \item Find two orthonormal bases for $X$ that contain
    the vector $e_1/||e_1||$, where $e_1=(1,0)$.
  \item Find two orthonormal bases for $X$ that contain
    the vector $e_2/||e_2||$, where $e_2=(0,1)$.
  \item Find an vector $v_2$ orthogonal to $v_1=(1,1)$.
  \item Find a matrix $H$ that is self-adjoint with respect to $(\;,\;)$, but
    \emph{not} with respect to $\<\;,\;\>$. \\
  \end{enumerate}

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

\item Let $H,M\colon X\to X$ be self-adjoint mappings, and $M$
  positive. 
  \begin{enumerate}
  \item Formulate and prove a necessary and sufficient condition for
    $M^{-1}H$ to be self-adjoint with respect to the standard inner
    product.
  \item Prove that $M^{-1}H$ is self-adjoint with respect to the inner
    product $\<x,y\>=(x,My)$. Conclude that there exists a basis
    $v_1,\dots,v_n$ of $X$ and $\mu_1,\dots,\mu_n\in\R$ such that
    \[
    Hv_i=\mu_iM v_i,\qquad \<v_i,v_j\>=\delta_{ij}.
    %(v_i,Mv_j)=\left\{\begin{array}{cc}1&i=j\\0&i\neq j\end{array}\right.
    \]
  \item Find formulas for $\<v_i,v_j\>$ and $\<v_i,M^{-1}Hv_j\>$ in
    terms of the standard inner product.
    \item Show that if $H$ has only positive eigenvalues, then so does $M^{-1}H$. 
  \end{enumerate}

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

\item Let $H,M\colon X\to X$ be self-adjoint mappings, $M$
  positive, and define
  $\displaystyle
  R_{H,M}(x)=\frac{(x,Hx)}{(x,Mx)}.
  $
  \begin{enumerate}
  \item Show that $\mu_1:=\min\{R_{H,M}(x)\mid x\in X\}$ exists,
    and write an equation relating $v_1,\mu_1$, $H$, and $M$. 
  \item Show that there is some $v_2\in X$ solving the
    constrained minimum problem
    \[
    \mu_2:=\min\big\{R_{H,M}(x)\mid (x,Mv_1)=0\big\}.
    \]
    Write an equation relating $v_2,\mu_2$, $H$, and $M$. 
  \item Find an invertible $U$ and diagonal $D$ such that $U^*MU=I$ and
    $U^*HU=D$.
  \item Characterize the diagonal entries of $D$ by a min-max
    principle.
  \end{enumerate}

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

\end{enumerate}

\end{document}







  
