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\begin{document}
\noindent
{\bf Math 408 Advanced Linear Algebra} \qquad Homework 8 {\hfill Your Name}

\medskip\centerline{
Eight points for each question.}

\begin{enumerate}
\item Suppose $\|\cdot\|$ is a matrix/algebra norm on $M_n$, and suppose $A \in M_n$.

(a) Let $x$ be an eigenvector of an eigenvalue $\mu$ of $A$, and let
$X \in M_n$ be such that every 
column of $X$ equals $x$. Show that 
$AX = \mu X$.

(b) Deduce from (a) that $r(A) \le \|A\|$.

\item A norm $\nu$ on $\IF^n$ is compatible with a norm $\|\cdot|$ on $M_n$
if $\nu(Ax) \le \|A\| \nu(x)$ for all $x \in \IF^n$.
Suppose $\|\cdot\|$ is an algebra norm on $M_n$. Show that 
the norm $\nu(x) = \|[x \cdots x]\|$ on $\IF^n$ is compatible with $\|\cdot\|$.


\item 
Show that the following conditions are equivalent
for any unitarily invariant norm $\|\cdot\|$ on $M_n$.

(a) $\|\cdot\|$ is an algebra norm.

(b) $\|E_{11}\| \ge 1$.

(c) $\|A\| \ge s_1(A)$ for any $A \in M_n$.

[Hint: Note that for any $A, B \in M_n$, $\|A\| \ge \|s_1(A)E_{11}\|$, and 
$\|AB\| \le \|s_1(A) B\|.$]


{\bf Remark} From the above result, one sees that 
for any unitarily invariant norm $\|\cdot\|$ on $M_n$, 
$\xi \|\cdot\|$ is an algebra UI norm if and only if $\xi \ge 1/\|E_{11}\|$.


\item (a) Let $n \ge m \ge 0$. Show that there is $A \in M_n$
such that $\lim_{k\rightarrow \infty} A^k$ exists
and has rank $m$.

(b) Give an example of $B \in M_n$ such that
$\lim_{k\rightarrow \infty} s_1(B^{k}) = 1$
but $\lim_{k\rightarrow \infty} B^k$ does not exist.


\item Suppose $\|x\|$ is a norm on 
a complex linear space $V$ satisfying the parallelogram identity
$$\|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2 + \|y\|^2) \quad \hbox{ for all } x, y \in V.$$
Define
$\langle x, y\rangle = a+ib$ with
$2a = \|x+y\|^2 - \|x\|^2 - \|y\|^2, 2b = \|x+iy\|^2 - \|x\|^2 - \|y\|^2$
for any $x, y \in V$.

(a) Show that $\langle x, y \rangle$ is indeed an inner product.

(b) Show that $\|x\| = \langle x, x\rangle^{1/2}$.

\item Suppose $A \in M_n$. 

(a) Show that there is a rank one matrix $T$ with 
$s_1(T) = s_n(A)$ such that $A-T$ is singular.

(b) Show that if $R \in M_n$ satisfies $s_n(R) > s_1(A)$, then 
$A-R$ is invertible.

\item 
Suppose $A\in M_n$ has eigenvalues $\lambda_1, \dots, \lambda_n$ with real parts
$a_1 \ge \cdots \ge a_n$. 
Show that for any $\xi > a_1$
then there is a Hermitian matrix $B \in M_n$ with $\lambda_1(B) < \xi$ 
such that $A-B$ has all eigenvalues lying in $\{\mu\in \IC: \mu+\bar \mu < 0\}$.



\item (extra credits)
(a) Show that the numerical radius is not an algebra norm.

(b) Show that 
the norm defined by $\|A\| = \xi w(A)$ is an algebra norm if and only if
$\xi \ge 4$.

\item (extra credits) Suppose $c = (c_1, \dots, c_n)^t$ with 
$c_1 \ge \cdots \ge c_n \ge 0$, and $x \in \IC^n$.
Show that 
$$\nu_c(x) = \max\{ |c^tPx|: P \in GP_n\} = \sum_{j=1}^n c_j Qx$$
for 
some $Q \in GP_n$ such that $Q x$ has nonnegative entries arranged in descending order.

Recall that $GP_n$ is the set of matrices that can be written as the product
of a permutation matrix and a diagonal unitary matrix.
Hint: Let $Q \in GP_n$ such that $\nu_c(x) = |c^tQx|$. 
Show that we may assume $Qx$ has nonnegative entries. Then show 
$Qx = (x_1, \dots, x_n)^t$ satisfies $x_i \ge x_{i+1}$ for any $i = 1, \dots, n-1$.


\item (extra credits) Recall that for any A$ \in M_n$, 
$W(aI + bA) = \{a + b\mu: \mu \in W(A)\} = a + bW(A)$. 
Also, $W(A) = W(U^*AU)$ for any unitary $U \in M_n$. Thus,
for any $A \in M_2$, we may replace $A$ by $A_0 = U^*(aA + bI)U$
and assume that $A_0 = \begin{pmatrix} a & b \cr 0 & -a\end{pmatrix}$
with $a, b \ge 0$. It is known that 
if $b = 0$ then $W(A_0)$ is a line segment joining  
$a, -a$; if
$a = 0$, then $W(A_0)$ 
is a circular disk centered at $0$ with radius $|b|/2$.
Suppose $ab \ne 0$.

(a) Show that $A_0$ is unitarily similar to 
$H+iG$ with $H = \diag(\alpha, -\alpha)$ and
$G = \begin{pmatrix}0 &\beta \cr -\beta & 0\cr\end{pmatrix}$,

\quad where
$\alpha = \sqrt{a^2+b^2/4}$ and $\beta = b/2$.

(b) Show that 
$$W(A_0) = e^{ir}\{v^*Hv + iv^*Gv: v \in \IC^n, v^*v = 1\}
= e^{ir}\{ \alpha x+ i\beta y: x+iy \in W(B_0)\},$$

\quad where 
$B_0 = \begin{pmatrix} 1 & 1 \cr -1 & -1\end{pmatrix}$
has numerical range equal to $\{ \mu: |\mu| \le 1 \}$.

(c) Deduce from (b) that 
$$W(A_0) = e^{ir}\{x+iy: (x/\alpha)^2 + (y/\beta)^2 \le 1\}$$
is an elliptical disk with foci $\pm \sqrt{\alpha^2-\beta^2} =  \pm a$, and 
semi-minor axis of length $\beta = b/2$.


{\bf Remark} The above result is known as the elliptical range theorem
asserting that the numerical range of $A \in M_2$
is always an elliptical disk with its eigenvalues $\mu_1, \mu_2$
as foci and length of minor axis equal to 
$\sqrt{(\tr AA^*) - |\mu_1|^2 - |\mu_2|^2}$.

\item (extra credits) Suppose $\langle x, y \rangle$ is an inner product on 
$\IC^n$.  Let $P = (p_{ij}) \in M_n$ be such that 
$p_{ij} = \langle e_j, e_i \rangle$ for $1 \le i, j \le n$, 
where $\{e_1, \dots, e_n\}$ is the standard basis for $\IC^n$.
Show that $\langle x, y\rangle = y^*Px$ and $P$ is a positive definite matrix.

Conversely, suppose $Q \in M_n$ is a positive definite matrix.
Show that 
$\langle x, y \rangle = y^*Qx$ is an inner product on $\IC^n$.

 
\end{enumerate}


\end{document}



