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\begin{document}
\noindent
{\bf Math 408 Advanced Linear Algebra} \qquad Homework 1 {\hfill Your Name}

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

(a) Find the (complex) eigenvalues of $A$, and the corresponding eigenvectors.

(b) Determine a complex matrix $S$ such that $S^{-1}AS$ is in diagonal form.

(c) Find a formula for $A^k$ for $k = 1,2,3,\dots$.


\item (24 points) 
Suppose $A = SDS^{-1} \in M_n$ such that $D = \diag(\lambda_1, \dots, \lambda_n)$,
and suppose $S$ has columns $x_1, \dots, x_n$ and $S^{-1}$ has rows $y_1^t, \dots, y_n^t$.

(a) Show that $A^k = SD^kS^{-1}$.

(b) Show that 
$A^k = \sum_{j=1}^n \lambda_j^k x_j y_j^t$ for every positive integer $k$.

(c) If $A$ is invertible, show that 
$A^k = SD^kS^{-1} =
\sum_{j=1}^n \lambda_j^k x_j y_j^t$ for every negative integer $k$.

(d) For any polynomial $f(z) = a_m z^m + \cdots + a_0$, let 
$f(A) = a_m A^m + \cdots + a_1A + a_0I_n$. 

\quad Show that 
$$f(A) = Sf(D)S^{-1} = \sum_{j=1}^n f(\lambda_j) x_j y_j^t.$$

(e) Show that $y_i^t x_j = \delta_{ij} = \begin{cases} 1 & \hbox{ if } i = j, \cr 
0 & \hbox{ if } i\ne j.\cr 
\end{cases}$

(f) Suppose $S$ has rows $u_1^t, \dots, u_n^t$ and $S^{-1}$ has columns
$v_1, \dots, v_n$. Show that 
$u_i^t v_j = \delta_{ij}$.

\item (12 points) Let $A\in M_n$ be 
diagonalizable with distinct eigenvalues $\lambda_1, \dots, \lambda_k$,
and let 
$$f(z) = (z-\lambda_1)\cdots (z-\lambda_k).$$

(a) Show that $A$ is similar to $\lambda_1 I_{n_1} \oplus \cdots \oplus \lambda_k I_{n_k}$
with $n_1 + \cdots + n_k = n$.

(b) Show that $f(A) = (A-\lambda_1 I)\cdots (A-\lambda_k I) = 0_n$.

(c) If $g(z) = \det(zI-A) = (z-\lambda_1)^{n_1} \cdots (z-\lambda_k)^{n_k}$,
show that $g(A) = 0_n$.

 
\item (12 points) Let $x(s) = (x_1(s), \dots, x_n(s))^t$ be such that $x_1(s), \dots, x_n(s)$
are differentiable functions.  
Suppose $x'(s) = (x_1'(s), \dots, x_n'(s))^t = A x(s)$, where $A = SDS^{-1}$ with
$D = \diag(\lambda_1, \dots, \lambda_n)$, and 
$y(s)  = (y_1(s), \dots, y_n(s))^t = S^{-1}x(s)$

(a) Show that $y'(s) = S^{-1} x'(s)$ so that $y'(s) = D y(s)$.

(b) Show that $y_i(s) = c_ie^{s\lambda_i}$ with $c_i = y_i(0)$ for $i = 1, \dots, n$.

(c) Show that $x(s) = Sy(s) = S(c_1 e^{s\lambda_1}, \dots, c_n e^{s\lambda_n})^t$.

\item (Extra 6 points) Suppose $A \in M_m$ is upper triangular and 
$B \in M_n$ is lower triangular, and $C \in M_{m,n}$.
Assume $X \in M_{m,n}$ satisfies $AX-XB = C$.
Show that for $1 \le j \le n$, 
$$Col_j(C) = A Col_j(X) - \sum_{i=1}^n b_{ij} Col_i(X),$$
where $Col_i(R)$ denotes the $i$th column of a matrix $R$.

\end{enumerate}

\end{document}
