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\fancyhead[L]{\textbf{MATH 510 Homework 6\qquad\qquad Name:}}
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{\small
\begin{itemize}
\item Homework 6 is due March 22 at the beginning of class.
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\begin{enumerate}[{\rm 1.}]
\item \textbf{(Corrected March 13)} Let $A\in M_n(\C)$. Let $a_1, a_2,\ldots, a_n\in \C^n$ be the columns of $A$. Prove \emph{Hadamard’s inequality}: $|\det A|\le \boldsymbol\prod_{i=1}^n \|a_i\|$ with equality if and only if $A$ has orthogonal columns or $a_i=0$ for some $i$. \emph{Hint:} Use the QR factorization of $A$.
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\item Let $A\in M_n(\R)$ be given. Explain why $A$ is symmetric if and only if $A$ is normal and all its eigenvalues are real.
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\item Let $A\in M_n(\R)$ be a real normal matrix. If $AA^T$ has $n$ distinct eigenvalues, show that $A$ is symmetric.
\item 
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\item Show that any real orthogonal matrix has determinant $1$ or $-1$.
\item Let $A\in M_n(\R)$ be a real orthogonal matrix such that $\det(A)=1$. Show that there is a continuous function $\gamma:[0,1]\to O(n)$ such that $\gamma(0)=A$ and $\gamma(1)=I_n$. (That such a function is continuous means that $\gamma(t)=[B_{ij}(t)]\in M_n(\R)$ for some continuous functions $B_{ij}(t)$.) (\emph{Hint:} Use the spectral theorem for real normal matrices.)
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\item Show that any $B\in M_n(\C)$ of the form $B = A^\ast A$, $A\in M_n(\C)$ may be written $B = LL^\ast$, where $L\in M_n(\C)$ is lower triangular and has real non-negative diagonal entries. Explain why this factorization is unique if $A$ is nonsingular.
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