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\fancyhead[L]{\textbf{MATH 510 Homework 5\qquad\qquad Name:}}
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{\small
\begin{itemize}
\item Homework 5 is due March 8 at the beginning of class.
\item \emph{Problem 5 corrected on March 3, 7pm}
\end{itemize}}

\begin{enumerate}[{\rm 1.}]
\item A permutation matrix is an $n\times n$ matrix in which each column and each row has exactly one non-zero entry, and that entry equals $1$. Show that the set of permutation matrices is a subgroup of the group $O(n)$ of real orthogonal matrices. (Subgroup here means a subset which is closed under multiplication, inverses, and contains identity matrix.)
\item Show that if $A\in M_n(\C)$ is similar to a unitary matrix, then $A=B^{-1}B^\ast$ for some nonsingular $B$.
\item Show that the set of matrices that are similar to unitary matrices is a proper subset of the set of matrices for which $A^{-1}$ is similar to $A^\ast$. Hint: Consider the matrix $\diag(2,\frac{1}{2})$.
\item Let $A,B$ be commuting matrices with eigenvalues $\alpha_1,\alpha_2,\ldots,\alpha_n$ and $\beta_1,\beta_2,\ldots,\beta_n$ respectively. If $p(t,s)$ is a polynomial in two variables, show that $p(A,B)$ has eigenvalues $p(\alpha_1,\beta_{i_1}), p(\alpha_2,\beta_{i_2}),\ldots,p(\alpha_n,\beta_{i_n})$ for some permutation $(i_1,i_2,\ldots,i_n)$ of $(1,2,\ldots,n)$.
\item Let $A\in M_n(\C)$ be a matrix such that $\Tr(A^k)=0$ for all $k\ge 1$. Show that $A$ \textbf{is nilpotent}. \emph{Hint:} You may use that the elementary symmetric polynomials 
\[e_d=\sum_{1\le i_1<i_2<\ldots<i_d\le n} \lambda_{i_1}\lambda_{i_2}\cdots \lambda_{i_d}\]
are polynomials in the power sums $p_d=\sum_{i=1}^n \lambda_i^d$.
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