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\begin{document}

\title{MATH 351, Fall 2017, Homework \#10}

\author{\underline{\bf Due Friday, December 8}}



\maketitle


\begin{problem}[8.3, p. 452] Show for all $\vec x$ in $\mathbb R^n$ that the following hold:
\end{problem}

\begin{itemize}
\item[a)] $\|\vec x\|_\infty\leq \|\vec x\|$

\item[b)] $\|\vec x\|\leq \sqrt n\|\vec x\|_\infty$

\item[c)] $\|\vec x\|_1\leq \sqrt n \|\vec x\|$

\end{itemize}


\begin{problem}[8.10, p. 452] Show that for an $n\times n$ orthogonal matrix $A$ that ${\rm cond}(A)\leq n$.
\end{problem}

\begin{problem} Let $A$ be a symmetrix real $n\times n$ matrix wth eigenvalues $|\lambda_1| > |\lambda_2|\geq |\lambda_3|\geq \dotsb \geq |\lambda_k|\geq 0$. Let $W$ be the orthogonal complement of the span of the $\lambda_1$ eigenvectors. Show that $|\lambda_2|$ is the maximum of the ratios $$\frac{\|A\vec x\|}{\|\vec x\|}$$ where $\vec x$ is a non zero vector belonging to $W$.
\end{problem}

\begin{problem} Let $A$ be an $n\times n$ real matrix with non-negative entries such that the entries in each row sum up to $1$. Show that $1$ is an eigenvalue of $A$. What about if the entries in the columns sum to $1$ instead?
\end{problem}


\begin{problem} Show that if $A$ and $B$ are both invertible $n\times n$ real matrices, then $AB$ and $BA$ have the same eigenvalues.
\end{problem}

\begin{problem}[5.31, p. 291] Let $A$ be an $n\times n$ real matrix with distinct eigenvalues $\lambda_1,\dotsc,\lambda_n$. Show that ${\rm det}(A) = \lambda_1\dotsb \lambda_n$.
\end{problem}

\begin{problem}[cf. 5.36, p. 291] Let $A$ be a real $n\times n$ matrix, and let $p(\lambda) = \det(A - \lambda I)$ be its characteristic polynomial. If $A$ is diagonalizable show that $p(A) = 0$.
\end{problem}


\begin{problem}[4.16, p. 259] Show that $$\det\begin{pmatrix*} 1 & 1 & 1\\ x & y & z\\ x^2 & y^2 & z^2\end{pmatrix*} = (x - y)(y - z)(z - x)$$
\end{problem}


\begin{problem}[4.39, p. 269] Use Theorem 4.15 on p.\ 267 to compute the inverse of the matrix $$\begin{bmatrix} 5 & 2& 1\\ 2 & 2 & 2\\ 2 & 1 & 1\end{bmatrix}.$$
\end{problem}




\end{document}


