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%%%%%% Math 403 Homework \#3, Spring 2024 %%%%%%
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%%%%%% Instructor: Ezra Miller %%%%%%
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\title{\mbox{}\\[-8ex]Math 403 Homework \#3, Spring 2024\\\normalsize
Instructor: Ezra Miller\\[-2.5ex]}
\author{Solutions by: ...your name...\\[1ex]
Collaborators: ...list those with whom you worked on this assignment...\\
(1 point for each of up to 3 collaborators who also list you)}
\date{Due: 11:59pm Saturday 2 March 2024}
\maketitle
\vspace{-5ex}\collab{}%
\noindent
\textsc{Reading assignments}\total{}{42}%covers Lectures 13 - 16
\begin{enumerate}[itemsep=-1ex]\setcounter{enumi}{12}
\item%13.
for Thu.~22 February
\begin{itemize}
\item{}%
[Treil, \S 6.3.3--\S 6.3.4] singular value decomposition
\item{}%
[Lax, Chap.7: \S Norm of a Linear Map, \S Spectral Radius]
\item{}%
[Lax, Chap.8: \S Norm and Eigenvalues]
\item{}%
[Treil, \S 6.4] Applications of SVD: spectral radius, operator norm,
condition number
\end{itemize}
\item%14.
for Tue.~27 February
\begin{itemize}
\item{}%
[Stewart--Sun, \S II.2.1] matrix norms, consistency
\item{}%
[Stewart--Sun, \S IV.1 through Thm 1.3] general perturbation theorems
\end{itemize}
\item%15.
for Thu.~29 February
\begin{itemize}
\item{}%
[Stewart--Sun, \S IV.1.2 through Thm 1.6] Bauer--Fike theorem
\item{}%
[Lax, Appendix 7] Gershgorin's Theorem
\item{}%
[Stewart--Sun, \S IV.2 through \S IV.2.1] Gerschgorin theory
\end{itemize}
\item[16--17.]\addtocounter{enumi}{2}%16. and 17.
for Tue.~5 March and Thu.~7 March
\begin{itemize}
\item{}%
[Tapp, Chap.5] Lie algebras as tangent spaces to the identity
\item{}%
[Lax, Chap.9: Matrix-valued Functions, through \S Matrix Exponential]
\item{}%
[Wikipedia, \emph{Exponential map (Lie theory)}]
\end{itemize}
\end{enumerate}
\noindent
\textsc{Exercises}% covers Lectures 9-13 + a bit of 14
\begin{enumerate}
\item\ \score{}%1.
Prove that a subset of a manifold is open if and only if its preimage
under every chart is an open subset of a vector space. (Note: This
can be used to specify the topology on a manifold without
% appealing to a general notion of topological space
knowing what a topological space is, since vector spaces have the
usual
% topology
notion of ``open'': what it means for a neighborhood of a point in a
manifold to be open is well defined independent of which charts are
used~to~verify~openness.)
\item\ \score{}%2.
Prove that the sphere $S^2 = \big\{\xx \in \RR^3 \,\big|\, ||\xx|| =
1\big\}$ is a smooth manifold. Is it a rational algebraic variety
over the field~$\RR$?
\item\ \score{}%3.
Fix a field~$F$. Prove that the unipotent lower-triangular matrices
$U^- \subseteq \GL_n(F)$ map injectively to the set $\FL(F)$ of
complete flags in~$F^n$ expressed as the set of orbits of the group
$B^+$ acting on the right of~$\GL_n(F)$; that is, $U^- \into \FL(F) =
\GL_n(F)/B^+$.
\item\ \score{}%4.
Let $S_n \subseteq \GL_n(F)$ be the set of permutation matrices.
Prove that $\FL(F)$ is a rational algebraic variety with atlas
$\{\pi_w: wU^- \to \FL(F) \mid w \in S_n\}$ naturally indexed
by~$S_n$. You will need to use that any level set
% (actually, you should only need the zero level set)
of a polynomial with coefficients in~$F$ is closed.
\item\ \score{}%5.
The \emph{standard complex structure} on $\RR^{2n}$ is the
block-diagonal $2n \times 2n$ matrix $J_{2n}$ whose diagonal blocks
are all $\twobytwo{0&\!-1\\1&\ 0}$. Prove that $A \in \GL_{2n}(\RR)$
is complex-linear if and only if $A$ commutes with the standard
complex structure: $AJ_{2n} = J_{2n}A$.
\item\ \score{}%6.
Prove that if $\lambda \in \CC$ is an eigenvalue of a unitary matrix
then $|\lambda| = 1$.
\item\ \score{}%7.
Show that the standard Hermitian inner product on $\CC^n$ defines a
distance $d(\xx,\yy) = ||\xx-\yy||$ on~$\CC^n$. Give an example of an
isometry $\varphi$ of~$\CC^n$ such that $\varphi(\0) = \0$ but
$\varphi$ is not $\CC$-linear.
\item\ \score{}%8.
Let $\varphi$ be an orthogonal transformation of a real inner product
space~$V$. Assume that $\varphi^2 = -I$. Show that $\dim V$ is even,
say~$2n$. Moreover, prove that there exists a dimension~$n$ subspace
$W \subset V$ and an
% orthogonal transformation
isometry $\psi: W \to W^\perp$ such that, in the decomposition $V = W
\oplus W^\perp$, the operator $\varphi$ is given by the block matrix
$$%
\twobytwo{ \0 & -\psi^*
\\ \psi & \0 }
$$
(N.B. The result means that $\varphi$ can be thought of as
multiplication by~$i$ on a complex vector space whose real and
imaginary parts are $W$ and~$W^\perp$.)
\item\ \score{}%9.
True or false: the sum of two normal operators is normal. Justify.
\item\ \score{}%10.
Show that the space of positive (semi)definite real symmetric
matrices is convex. Is the same true with ``complex Hermitian'' in
place of ``real symmetric''?
\item\ \score{}%11.
Orthogonally diagonalize the matrix $A = \twobytwo{3&2\\2&3}$. Find
all square roots of~$A$; note that they are all self-adjoint.
\item\ \score{}%12.
Find a singular decomposition of the matrix $A = \twobytwo{2&3\\0&2}$.
Use it to find $\max_{\Vert\xx\Vert \leq 1} \Vert A\xx\Vert$ and
$\min_{\Vert\xx\Vert = 1} \Vert A\xx\Vert$, as well as the vectors
where this maximum and minimum are attained. Describe geometrically
the image under $A$ of the closed unit disk in~$\RR^2$.
\item\ \score{}%13.
Prove that the operator norm of a matrix~$A$ coincides with the
Frobenius norm of~$A$ if and only if $A$ has rank at most~$1$.
\end{enumerate}
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