
\documentclass[11pt]{article}
%\baselineskip 16pt
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphicx}

\setlength{\oddsidemargin}{-0.15in}
\setlength{\topmargin}{-0.5in}
\setlength{\textheight}{9in}
\setlength{\textwidth}{6.5in}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\def\qed{\hfill $\Box$\medskip}



\def\IC{{\mathbb C}}
\def\ID{{\mathbb D}}
\def\IR{{\mathbb R}}
\def\IZ{{\mathbb Z}}
\def\ZZ{{\IZ}}
\def\IF{{\mathbb F}}
\def\IQ{{\mathbb Q}}
\def\IN{{\mathbb N}}
\def\tr{{\mathrm tr}\,}
\def\Sp{{\mathrm Sp}\,}
\def\Ev{{\mathrm Ev}\,}
\def\rank{{\rm rank}\,}
\def\conv{{\rm conv}\,}
\def\diag{{\rm diag}\,}
\def\bD{{\bf D}}
\def\bW{{\bf W}}
\def\ba{{\bf a}}
\def\cS{{\mathcal S}}
\def\cT{{\mathcal T}}
\def\cN{{\mathcal N}}
\def\cU{{\mathcal U}}
\def\cM{{\mathcal M}}
\def\cH{{\mathcal H}}
\def\cR{{\mathcal R}}
\def\cH{{\mathcal H}}
\def\cF{{\mathcal F}}
\def\cL{{\mathcal L}}
\def\cP{{\mathcal P}}
\def\cV{{\mathcal V}}
\def\cK{{\mathcal K}}
\def\BH{{\cB(\cH)}}
\def\SH{{\cS(\cH)}}
\def\FH{{\cF(\cH)}}
\def\KH{{\cK(\cH)}}
\def\re{{\rm Re}\,}
\def\im{{\rm Im}\,}
\def\cl{{\bf cl}\,}
\def\span{{\rm span}\,}
\def\la{{\langle}}
\def\ra{{\rangle}}
\def\vec{{\rm vec}}

\def\bV{{\mathbf V}}
\def\bP{{\mathbf P}}
\def\bA{{\mathbf A}}
\def\bB{{\mathbf B}}
\def\bF{{\mathbf F}}
\def\bmu{{mathbf \mu}}
\def\red{\color{red}}
\def\blue{\color{blue}}
\def\green{\color{green}}
\def\pink{\color{pink}}
\def\grey{\color{dark}}


\begin{document}
\noindent
{\bf Math 408 Advanced Linear Algebra} \qquad Homework 4 {\hfill Your Name}



\begin{enumerate}
\item (12 points) Let $A = \begin{pmatrix} 1 & 1-i & i\cr 2 & 0 & 0\cr\end{pmatrix}$. 

(a) Find the nonzero singular values $s_1\ge s_2 > 0$ of $A$.

(b) Find the singular value decomposition of $A$, 

\quad i.e., find unit vectors 
$u_1, u_2 \in \IC^2$ and $v_1, v_2 \in \IC^2$ such that $A = s_1 u_1v_1^* + s_2 u_2v_2^*$.

(c) Find a positive definite $P \in M_2$ and $V \in M_{2,3}$ with $VV^* = I_2$ 
such that $A  = PV$.

(d) Find a positive definite $Q \in M_3$ and $U \in M_{2,3}$ with $UU^* = I_2$ such that
$A = UQ$.

(e) Find a unitary $U \in M_2, V \in M_3$ so that $U^*AV = s_1 E_{11} + s_2 E_{22}\in M_{2,3}$.

Hint: For (a) and (b),  find the eigenvalues and eigenvectors  of $AA^*$.

\item (4 points) Show that a matrix $A \in M_n$ is positive semidefinite if and only if 
its singular values are the same as its eigenvalues.

\item (4 points) Let $A \in M_{m,n}$.
Show that the number of nonzero singular values of $A$ equals:
$$\rank(A) = \rank(AA^*) = \rank(A^*A).$$

\item (4 points)
Let $A, B \in M_{m,n}$. Show that there exist unitary $U \in M_m$ and $V \in M_n$
such that $A = UBV$ if and only if $A$ and $B$ have the same singular values.

\item (8 points) Let $A = A^t \in M_n$ be a complex symmetric matrix. 
Suppose $u \in \IC^n$ is a unit vector such that $|u^tAu|$ is maximum among all 
unit vectors.

(a) Show that there is a unit vector $v$ such that $v^t A v = \ |u^t Au|$.

(b) Suppose $U$ is unitary with $v$ in part (a) as the first column.
Show that $U^t A U = [v^t Av] \oplus A_1$ 

\qquad such that $A_1 = A_1^t \in M_{n-1}$.

(c) Show by induction that is a unitary $W \in M_n$ such that 
$W^tAW = \diag(s_1, \dots, s_n)$.

(d) Show that $s_1, \dots, s_n$ in (c) are the singular values of $A$.

Hint for (b): Suppose $a_{1j} = |a_{1j}| e^{ir} \ne 0$ for some $j > 1$. 
Let $x_\theta = \cos \theta e_1 + \sin \theta e_j e^{-ir}$.
Show that $f'(0) > 0$ if
$$f(\theta) = |x_\theta^t (U^tAU) x_\theta|^2 = 
(a \cos^2\theta + 2 |a_{1j}| \sin\theta\cos\theta 
+ c_1 \sin^2\theta)^2 + (c_2 \sin^2\theta)^2,$$
where $a_{jj} e^{-i2r} = c_1 + ic_2$ with  $c_1, c_2 \in \IR$. 
Then deduce that for sufficiently small $\theta > 0$, 
$f(\theta) > f(0) = |v^tAv|^2$ so that  
$|v_\theta^t A v_\theta| > |v^tAv|$ for the unit vector $v_\theta = Ux_\theta$.

\item (6 points)
Let $A \in M_n$ be positive semidefinite with 
eigenvalues $\lambda_1 \ge \cdots \ge \lambda_n \ge 0$.

(a) Show that $\lambda_1  = \max\{ x^*Ax: x\in \IC^n, x^*x = 1\}.$

(b) Show that $\lambda_n = \min\{x^*Ax: x\in \IC^n, x^*x = 1\}$.

Hint: Suppose $A = UDU^*$ with $D = \diag(\lambda_1, \dots, \lambda_n)U^*$. Then
$x^*Ax = y^* D y = \sum_{j=1}^n \lambda_j |y_j|^2$ if
$y = U^*x = (y_1, \dots, y_n)^t$. Argue that $\sum_{j=1}^n |y_j|^2 = 1$ so that
the maximum will occur when $|y_1|^2 = 1$ and minimum will occur when $|y_n|^2 = 1$.

\item (6 points)
Let $A \in  M_{m,n}$ with nonzero singular values $s_1 \ge \cdots \ge s_k > 0$.
Use the result in the last question or otherwise to show that 
$$s_1 = \max\{ \|Av\|: v \in \IC^n, \|v\| = 1\} \qquad \hbox{ and } \qquad
s_1 = \max\{ \|A^*u\|: u \in \IC^m, \|u\| = 1\}.$$
What can you can about the minimum values if $m, n > 1$?

Hint: Recall that $\|w\| = (w^*w)^{1/2}$ for a complex vector $w$
so that $\|Av\| = (v^*A^*Av)^{1/2}$. 


\item (Extra credit, 4 points) Suppose $A = -A^t$ is skew-symmetric. 
Suppose $u_1, u_2\in \IC^n$ are orthonormal pairs such that
$|u_1^t A u_2|$ is maximum, and $U$ is unitary with $u_1, u_2$ as the first
two columns. Show that
$U^tAU = \begin{pmatrix} 0 & a_{12} \cr -a_{12} & 0 \cr \end{pmatrix}
\oplus A_1$ 
for some $A_1 = -A_1^t \in M_{n-2}$.   

Hint:  Let $U^tAU = (a_{ij})$. If $a_{1j} \ne 0$, replace $u_2$ by
$v_\theta = U( \cos \theta \mu_2 e_2 + \sin \theta \mu_j e_j)$, where $a_{12}\mu_2 = |a_{12}|$
and $a_{1j} \mu_j = |a_{1j}|$.  Showo that $|u_1^t A  v_\theta| > |u_1^t A u_2|$
for sufficiently small $\theta > 0$.
If $a_{2j} \ne 0$, one can replace $u_1$ by $w_\theta = U(\cos \theta \xi_1 e_1 + 
\sin \theta   \xi_2 e_2)$ and show that $|u_1t^tAu_2|$ can be increased.
\end{enumerate}
\end{document}



