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\fancyhead[L]{\textbf{MATH 510 Homework 9\qquad\qquad Name:}}
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{\small
\begin{itemize}
\item Due April 26 at the beginning of class.
\end{itemize}}

\begin{enumerate}[{\rm 1.}]
\item Suppose that $A\in M_n$ is a real matrix whose $n$ Gershgorin disks are all mutually disjoint. Show that all eigenvalues of $A$ are real. 
\item Suppose that $A\in M_n$ is idempotent $(A^2=A)$ but $A\neq I$. Show that $A$ cannot be strictly diagonally dominant. ($A$ is called \emph{strictly diagonally dominant} if $\displaystyle |a_{ii}|> \sum_{\substack{1\le j\le n\\j\neq i}} |a_{ij}|$ for all $i=1,2,\ldots,n$.)
\item Let $A\in M_n(\{0,1\})$ be a matrix whose entries are either $0$ or $1$. Show that $A$ has property (SC) (equivalently, its directed graph $\Gamma(A)$ is strongly connected) if and only if every entry of the matrix $(I+A)^{n-1}$ is positive. (You may use that the $i,j$ entry of $A^k$ is the number of paths in $\Gamma(A)$ from $i$ to $j$ of length $k$.)
\item Let $A\in M_n$ be a skew-Hermitian matrix. For each $\varepsilon\in\R$, let $A_\varepsilon=\exp(\varepsilon A)$ and let $\lambda_\varepsilon$ be an eigenvalue of $A_\varepsilon$. Show that \[\limsup_{\varepsilon\to 0}\frac{1}{\varepsilon}|\lambda_\varepsilon-1|\le \|A\|_\infty,\]
where $\|A\|_\infty=\max_p \sum_j |a_{pj}|$ is the max-row-sum matrix norm.
\item What are necessary and sufficient conditions on the signs of its minors for a Hermitian matrix $A$ to be negative definite (semidefinite)? (\emph{Hint:} This is related to Sylvester's Criterion.)
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