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\begin{document}
\title{Math 200b Winter 2021 Homework 3}
\date{Due 1/29/2020 by midnight on Gradescope}
\maketitle


1.  Let $F$ be a field.  Prove that if $n \leq 3$, two matrices $A, B \in M_n(F)$ are similar if and only if $\on{charpoly}(A) = \on{charpoly}(B)$ and $\on{minpoly}(A) = \on{minpoly}(B)$.  Give an example to show this result does not hold for matrices in $M_4(F)$ in general.

\bigskip

2.  Let $F$ be an algebraically closed field.  Consider the matrix
\[
M = \begin{pmatrix} 0 & -1 & -1 \\ 0 & 0 & 0 \\ -1 & 0 &0 \end{pmatrix} \in M_3(F).
\]
Find the minimal and characteristic polynomials of $M$, and the rational and Jordan canonical forms
of $M$.  (The answers may depend on the characteristic of $F$.)

\bigskip

3.  Consider the three matrices 
\[
A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \  
B = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \ \text{and}\
C = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix}.
\]
Are any of these matrices similar to each other over $\mb{C}$?  Justify your answer.


\bigskip

4.  let $F$ be a field.   Find representatives of each of the similarity classes of matrices $A \in \operatorname{GL}_2(F)$ such that $A$ has multiplicative order exactly $4$ in this group, and thus calculate exactly how many such similarity classes there are, when 

\medskip

(a) $F = \mb{Q}$.

\medskip

(b) $F = \mb{C}$.

\medskip

(c) $F$ is a field of characteristic $2$.



\bigskip


\newpage

5.  Let $F$ be a field.  Let $A \in M_n(F)$ and let $f = \on{charpoly}(A)$ and $g = \on{minpoly}(A)$.   Let $A^t$ be the transpose of $A$.  

(a)   Prove that $\on{charpoly}(A^t) = f$ and $\on{minpoly}(A^t) = g$.


(b)  Prove that $f = g$ if and only if $A$ is similar to the companion matrix $C_f$.

(c)  Show that $(C_h)^t$ is similar to $C_h$ for any companion matrix $C_h$.

(d)  Show that $A$ is similar to $A^t$. 

\bigskip


6. Let  $J \in M_n(\mb{C})$ be a single Jordan block corresponding to the eigenvalue $\lambda \in \mb{C}$.  

(a)  If $\lambda \neq 0$, prove that the Jordan canonical form of $J^2$ is also a single Jordan block, with eigenvalue $\lambda^2$.

(b)  If $\lambda = 0$, prove that the Jordan form of $J^2$ consists of two Jordan blocks, of size $n/2$ and $n/2$ 
if $n$ is even and of size $(n+1)/2$ and $(n-1)/2$ if $n$ is odd.

(c)  Determine necessary and sufficient conditions for a matrix $M \in M_n(\mb{C})$ to have a square root, that is for there to exist $N \in M_n(\mb{C})$ such that $N^2 = M$.

\bigskip


\end{document}
