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\begin{document}
\noindent
{\bf Math 408 Advanced Linear Algebra} \qquad Examination 1 {\hfill Due: noon, 7 October.}



\begin{enumerate}

\item Let $T = \begin{pmatrix} A & B \cr C & D \cr \end{pmatrix} \in M_n$ be 
such that $A \in M_k$ is invertible.

(a) Show that $\begin{pmatrix} I_k & 0 \cr -CA^{-1} & I_{n-k}\cr\end{pmatrix} 
\begin{pmatrix} A & B \cr C & D \cr \end{pmatrix} 
\begin{pmatrix} I_k & -A^{-1}B \cr 0  & I_{n-k}\cr\end{pmatrix} = 
\begin{pmatrix} A & 0 \cr 0 & D -CA^{-1}B \cr\end{pmatrix},$
and 
$$\det(T) = \det(A)\det(D-CA^{-1}B).$$

(b) Show that  $T$ is invertible if and only if $D-CA^{-1}B$ is invertible, and 
$$T^{-1} =   \begin{pmatrix} I_k & -A^{-1}B \cr 0  & I_{n-k}\cr\end{pmatrix}
\begin{pmatrix} A^{-1} & 0 \cr 0 & (D - CA^{-1}B)^{-1}  \cr\end{pmatrix}
\begin{pmatrix} I_k & 0 \cr -CA^{-1} & I_{n-k}\cr\end{pmatrix}.$$

\item Let $A \in M_7$ with minimal polynomial $(x-1)^2(x-i)^3$. Determine all 
possible Jordan forms of $A$.

\item (a) Let $A = J_k(\lambda)$ with $k \ge 2$ and  $\lambda = \mu^2 \ne 0$. 
Show that there is an invertible matrix 
$S\in M_k$ such that $S^{-1}AS = \lambda I_k + 2\mu  N_k + N_k^2 = (\mu I_k + N_k)^2$,
and hence $A = B^2$ for $B = S(\mu I_k + N_k)S^{-1}$.

(b) Show that for every invertible matrix $T \in M_n$, there is $R \in M_n$
such that $T = R^2$.

\item Suppose $B = J_n(0)$. Then $B^2$ has Jordan form 
$$\begin{cases} J_{k}(0) \oplus J_{k}(0) & \hbox{  if } n = 2k,\cr
J_{k}(0) \oplus J_{k+1}(0) & \hbox{ if } n=2k+1.\cr\end{cases}.$$
Deduce that there is no matrix $T\in M_6$ such that $T^2 = J_2(0) \oplus J_4(0)$.

\item Show that $A, B \in M_2$ are unitarily similar if and only if $\tr A = \tr B, \tr A^2 = \tr B^2,
\tr AA^*  = \tr BB^*$.

Hint: To prove the sufficiency, 
first show that $A$ and $B$ have the same eigenvalues, and hence
by unitary similarities, the triangular forms $T_1$ and $T_2$ 
of $A$ and $B$ have the same
diagonal entries. Then show that the $(1,2)$ entries of the two triangular 
matrices can be changed to the same nonnegative number by applying diagonal unitary
similarities $D_1^*T_1D_1$ and $D_2^*T_2D_2$.

\item Let $w = e^{i2\pi/3}$, and 
$F = \frac{1}{\sqrt 3} \begin{pmatrix} 1 & 1 & 1 \cr 1 & w & w^2\cr
1 & w^2 & w \cr\end{pmatrix}$.

(a) Show that $F$ is unitary. 

(b) Suppose $P = E_{12} + E_{23} + E_{31}\in M_3$. Show that the $PF = FD$ with
$D = \diag(1, w, w^2)$.

(c) Suppose $A = a_1I + a_2 P + a_3P^2 = \begin{pmatrix} a_1 & a_2 & a_3 \cr 
a_3 & a_1 & a_2 \cr a_2 & a_3 & a_1 \cr\end{pmatrix}$. Determine 
$F^*AF$.

\item Suppose $B \in M_n$ is invertible. Show that $B = LV$
for some unitary $V$ and lower triangular matrix $L$ in $M_n$.
Deduce that every invertible positive semidefinite matrix $A$ can be written as 
$L_1 L_1^*$ such that $L_1$ is lower triangular.

\item 
(a) Show that $A\in M_n$ is normal if and only if $\|Av\| = \|A^*v\|$ for all $v \in \IC^n$.

(b) Show that $A\in M_n$ is skew-symmetric if and only if $u^tAu = 0$ for all $u \in \IC^n$.

Hint: For the sufficiency, 
write $A = S + K$ with $S = (A+A^t)/2$ is symmetric
and $K = (A-A^t)/2$ is skew-symmetric. Show that $S = 0$.

\end{enumerate}
\end{document}



