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% Homework for the course "Math 554: Linear Analysis",
% Autumn quarter, 2008, Anne Greenbaum.
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{\scriptsize Math 554, Autumn 2008} \hfill
\begin{center}
\large
Assignment 4. Due Monday, Oct. 27.
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\begin{tabular}{ll}
Reading: & Class Notes, pp. 52--60 \\
& Chapter 2 in Horn and Johnson.
\end{tabular}
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% The questions!
\begin{enumerate}
\item
\begin{description}
\item[(a)]
Let $A \in {\bf C}^{n \times n}$ be normal and suppose all eigenvalues
of $A$ are real. Show that $A$ is Hermitian.
\item[(b)]
Let $A \in {\bf C}^{n \times n}$ be normal and suppose all eigenvalues
of $A$ have absolute value $1$. Show that $A$ is unitary.
\item[(c)]
Show by examples that (a) and (b) both fail if the normality assumption
is replaced by the weaker assumption that $A$ is diagonalizable.
\end{description}
\item
Use the Schur Triangularization Theorem to show that every matrix
$A \in {\bf C}^{n \times n}$ is ``almost'' diagonalizable in the
following two senses:
\begin{description}
\item[(a)]
Given $\epsilon > 0$, there is a matrix $\tilde{A} \in {\bf C}^{n \times n}$
with distinct eigenvalues for which $\| A - \tilde{A} \|_F < \epsilon$.
\item[(b)]
Given $\epsilon > 0$, there is an upper triangular matrix $T$
similar to $A$ for which $| t_{ij} | < \epsilon$ for all $i < j$.
\end{description}
\item
\begin{description}
\item[(a)] What are the possible Jordan canonical forms of $A$ if
$p_A (t) = ( t + 4 )^3 ( t - 2 )^2$?
\item[(b)]
Find the Jordan canonical form for
\[
A = \left( \begin{array}{ccc} 3 & 1 & 2 \\ 0 & 3 & 0 \\ 0 & 0 & 3
\end{array} \right) .
\]
\end{description}
\item
Let $A,B \in {\bf C}^{n \times n}$ be diagonalizable. Show that
$A$ and $B$ are {\em simultaneously diagonalizable} (that is,
there exists one invertible matrix $S$ for which both
$S^{-1} A S$ and $S^{-1} B S$ are diagonal) if and only if
$AB = BA$. Proceed as follows:
\begin{description}
\item[(a)] Show that if $A$ and $B$ are simultaneously diagonalizable
then $AB=BA$.
\item[(b)]
Suppose $AB = BA$. Let $\lambda_1 , \ldots , \lambda_k$ be the
distinct eigenvalues of $A$, with eigenspaces $E_1 , \ldots , E_k$
and associated projections $P_1 , \ldots , P_k$.
Show that $B E_i \subset E_i$ for each $i$, and deduce that
$B P_i = P_i B$ for each $i$.
\item[(c)]
Suppose $AB = BA$. Let $\{ v_1 , \ldots , v_n \}$ be a basis of
${\bf C}^n$ consisting of eigenvectors of $B$. Show that for each $i$,
$1 \leq i \leq k$, the vectors $\{ P_i v_1 , \ldots , P_i v_n \}$
span $E_i$, where $E_i$ and $P_i$ are as in part (b). Also show
that each nonzero vector $P_i v_j$ is an eigenvector of $B$.
\item[(d)]
Suppose $AB=BA$. Deduce that there is a basis of ${\bf C}^n$
consisting of vectors which are eigenvectors of both $A$ and $B$.
Conclude that $A$ and $B$ are simultaneously diagonalizable.
\end{description}
\item
Do Problem 1 on p. 76 of Horn and Johnson.
\item
Prove the real version of the Schur Theorem: If
$A \in {\bf R}^{n \times n}$, there is an orthogonal matrix
$V \in {\bf R}^{n \times n}$ such that
\[
V^T A V = \left( \begin{array}{ccc} A_1 & \ldots & \ast \\
& \ddots & \vdots \\
& & A_k
\end{array} \right) ,
\]
is {\em block} upper triangular,
where each diagonal block $A_i$ is either a real $1 \times 1$ matrix
or a real $2 \times 2$ matrix whose eigenvalues are a complex
conjugate pair $\lambda \neq \bar{\lambda}$.
\end{enumerate}
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