\documentclass[12pt]{amsart}
\usepackage{amsmath,amsthm,amsfonts,latexsym,amssymb}
\pagestyle{headings}
\setlength{\textwidth}{36true pc}
\setlength{\headheight}{8true pt} % needed for 12pt documents
\setlength{\oddsidemargin}{15 truept}
\setlength{\evensidemargin}{15 truept}
\setlength{\textheight}{572true pt}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lm}[thm]{Lemma}
\newtheorem{prp}[thm]{Proposition}
\newtheorem{clm}[thm]{Claim}
%
\theoremstyle{definition}
\newtheorem{df}[thm]{Definition}
\newtheorem{exmp}[thm]{Example}
\newtheorem{prb}[thm]{Problem}
%
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
%
\numberwithin{equation}{section}
%
\newcommand{\thref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lmref}[1]{Lemma~\ref{#1}}
%
\DeclareMathOperator{\satf}{\models}
\DeclareMathOperator{\miff}{\Longleftrightarrow} 
                                           % math `if and only if'
\DeclareMathOperator{\mimplies}{\Longrightarrow} 
                                           % math `implies'
\DeclareMathOperator{\mimplied}{\Longleftarrow}  
                                           % math `implied by'
\newcommand{\set}[2]{\{#1\,:\,\text{#2}\}} % { x : ... }
\DeclareMathOperator{\id}{\textsl{id}}     % The identity map
\DeclareMathOperator{\miffdef}{\stackrel{def}{\miff}}
                                           % defining iff
\DeclareMathOperator{\eqdef}{\stackrel{def}{=}}
                                           % defining equation

\DeclareMathOperator{\neareq}{\stackrel{\ae}{=}}
\DeclareMathOperator{\aeqv}{\stackrel{ae}{\equiv}}
\DeclareMathOperator{\aeq}{\stackrel{ae}{=}}
%
\DeclareMathOperator{\BZ}{\mathbb{Z}} % integers
\DeclareMathOperator{\BQ}{\mathbb{Q}} % rationals
\DeclareMathOperator{\BR}{\mathbb{R}} % reals
\DeclareMathOperator{\BC}{\mathbb{C}} % complex numbers
\DeclareMathOperator{\BE}{\mathbb{E}} % 
\DeclareMathOperator{\BF}{\mathbb{F}} % 
\DeclareMathOperator{\BK}{\mathbb{K}} % 
\DeclareMathOperator{\BL}{\mathbb{L}} % 
\DeclareMathOperator{\re}{Re}         % real part
\DeclareMathOperator{\im}{Im}         % imaginary part
\DeclareMathOperator{\Ker}{Ker}
%
\DeclareMathOperator{\meet}{\wedge}
\DeclareMathOperator{\join}{\vee}
\DeclareMathOperator{\union}{\cup}
\DeclareMathOperator{\inter}{\cap}  % intersection
\DeclareMathOperator{\iso}{\cong}   % isomorphic
\DeclareMathOperator{\piso}{\simeq} % polynomially isomorphic
\newcommand{\Star}{^ *}
\DeclareMathOperator{\compose}{{\scriptstyle \circ}} % f o g
%
\newcommand{\wec}[1]{{\mathbf{#1}}}  % notation for vectors in algebras
%\newcommand{\wec}[1]{{\bar{#1}}} % which one do you prefer?
\newcommand{\wrel}[1]{\;#1\;}     % a\wrel{\alpha}b = a \alpha b
%              % this provides a flexible way to change the spacing
%
\newcommand{\m}[1]{{\mathbf{\uppercase{#1}}}}
\newcommand{\vr}[1]{{\mathcal{\uppercase {#1}}}}
%
\DeclareMathOperator{\Var}{\mathbf{V}}
\DeclareMathOperator{\Ho}{\mathbf{H}}
\DeclareMathOperator{\Su}{\mathbf{S}}
\DeclareMathOperator{\Pd}{\mathbf{P}}
\DeclareMathOperator{\Iso}{\mathbf{I}}
\DeclareMathOperator{\Ps}{\mathbf{P}_s}
\newcommand{\fin}[1]{{\mathbf{#1}}_{\text{\upshape fin}}}
%
\DeclareMathOperator{\Pol}{Pol}
\DeclareMathOperator{\Clo}{Clo}
\DeclareMathOperator{\Con}{Con}
\DeclareMathOperator{\Conb}{\mathbf{Con}} % Con(A) as a lattice
\DeclareMathOperator{\cg}{Cg}
\DeclareMathOperator{\sg}{Sg}
\newcommand{\Cg}[1]{{\cg}^{\m{#1}}}
\newcommand{\Sg}[1]{{\sg}^{\m{#1}}}
%
\newcommand{\M}[3]{\text{\upshape M}_{\m{#1}} (#2, #3)}
\newcommand{\minl}[1]{\text{\upshape Min}_{\m{#1}}}
%
\DeclareMathOperator{\typ}{typ}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\Sep}{Sep}
\DeclareMathOperator{\ssb}{ssb}
\DeclareMathOperator{\sst}{sst}
%
\DeclareMathOperator{\utyp}{\mathbf{1}}% unary      
\DeclareMathOperator{\atyp}{\mathbf{2}}% affine     
\DeclareMathOperator{\btyp}{\mathbf{3}}% boolean    
\DeclareMathOperator{\ltyp}{\mathbf{4}}% lattice    
\DeclareMathOperator{\styp}{\mathbf{5}}% semilattice
\DeclareMathOperator{\ityp}{\mathbf{i}}
\DeclareMathOperator{\jtyp}{\mathbf{j}}
%
\DeclareMathOperator{\solv}{\stackrel{s}{\sim}}
\DeclareMathOperator{\ssolv}{\stackrel{ss}{\sim}}
%
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}
%
\newcommand{\lb}{\langle} % <...>
\newcommand{\rb}{\rangle}
\newcommand{\lv}{{[\![}}  % [[...]]
\newcommand{\rv}{{]\!]}}
%
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
%
\newcommand{\vep}{\varepsilon}
\newcommand{\di}{\diamond}
%


\begin{document}

\noindent
{\large \bf Additional notes \#5.}

\bigskip

\noindent
{\bf Things that ``everyone knows'' about the Jordan form.}

\bigskip

The Jordan canonical form for $T\in {\rm End}_{\BF}(V)$,
where $V$ is an $n$-dimensional ${\BF}$-space, 
is an $n\times n$ block-diagonal matrix
where each block has the form 
$$
J_i = \left[\begin{array}{ccccc}
            \lambda & 1 & 0 & \cdots & 0\\
            0 & \lambda & 1 & \cdots & 0\\
            0 & 0 & \lambda & \cdots & 0\\
           \vdots & & & & \vdots\\
            0 & 0 & 0 & \cdots & \lambda\\
            \end{array}\right].
$$
(Under a different convention, the Jordan blocks 
have the form $J_i^t$ instead.)

The Jordan form is obtained by 
\begin{itemize}
\item considering $V$ as an ${\BF}[t]$-module,
\item expressing this module as
$$
{}_{{\BF}[t]}V\cong \left({\BF}[t]/(x-\lambda_1)^{e_1}\right)
\oplus\cdots \oplus
\left({\BF}[t]/(x-\lambda_k)^{e_k}\right), 
$$
\item and then showing the left multiplication
by $t$ on a direct summand  ${\BF}[t]/(x-\lambda)^{e}$
of the module on the right has a matrix that is an 
$e\times e$ Jordan matrix
with $\lambda$'s down the diagonal.
\end{itemize}

\bigskip

\begin{enumerate}
\item[(1)] The Jordan form can be determined from the 
elementary divisors of ${}_{{\BF}[t]}V$.
(Each elementary divisor $(x-\lambda)^e$ of
${}_{{\BF}[t]}V$ corresponds to an $e\times e$ Jordan
block with $\lambda$'s on the diagonal.)
\bigskip
\item[(2)] The Jordan form can be determined from the rational
canonical form. (Reason: the elementary divisors can be determined
by factoring the invariant factors into powers of 
distinct linear factors.)
\bigskip
\item[(3)] The Jordan form is unique up to a permutation
of the Jordan blocks. (Reason: the elementary divisors can be 
determined from the Jordan blocks, and the elementary divisors 
of a module over a PID are unique
up to a permutation of order.)
\bigskip
\item[(4)] The diagonal entries of the Jordan form of
$A$ are the eigenvalues of $A$, with the correct
multiplicity. The number of Jordan blocks corresponding
to a given eigenvalue $\lambda$ is the dimension of 
the eigenspace corresponding to $\lambda$.
(Reason: The Jordan form represents the same transformation
with respect to a new basis, so the eigenvalues and
characteristic polynomial of $A$ are the same as those
of the Jordan form for $A$. Thus, one only has to verify
the statement for matrices already in Jordan form,
and this is easy.)
\bigskip
\item[(5)] A matrix $A$ is ``diagonalizable over ${\BF}$''\footnote{% 
There exist invertible $S$ and diagonal
$D$ in $M_n({\BF})$ such that $S^{-1}AS = D$} if and
only if the Jordan form of $A$ is diagonal.
(Reason: A diagonal matrix is already in Jordan form,
so this result follows from the uniqueness of the
Jordan form.)
\bigskip
\item[(6)] A matrix $A$ is diagonalizable over $\BF$
if and only if its minimal polynomial 
factors into distinct linear factors over $\BF$.
(Reason: For the forward direction, notice that
the minimal polynomial of a diagonal matrix factors 
into distinct linear factors. For the reverse direction,
if $m_A(t)$ factors into distinct linear factors,
then all invariant factors also factor this way,
since they divide $m_A(t)$. But this forces all
elementary divisors to be linear, $(x-\lambda)^1$.
This forces all Jordan blocks to be $1\times 1$ blocks,
so the Jordan form is diagonal.)
\bigskip
\item[(7)] A family $\mathcal F$ of $n\times n$
matrices are simultaneously diagonalizable iff and only
if each matrix in $\mathcal F$ is diagonalizable 
and any two matrices in $\mathcal F$ commute.
(Reason: the two conditions for simultaneous
diagonalizability are necessary, since simultaneously
diagonalizable matrizes are diagonalizable, and
diagonal matrices commute. To see that the
conditions are sufficient, choose a basis
that diagonalizes the first matrix $A_1$.
With respect to this basis the others are block
diagonal matrices, where on each block $A_1$ is a scalar matrix.
%since $A_1$ commutes with the others. 
Now $A_2$
is block diagonal, and each of its blocks has a minimal
polynomial dividing the minimal polynomial of $A_2$.
By item (6), this shows that each block of
$A_2$ is diagonalizable. Thus we can choose
a basis that diagonalizes $A_2$
block by block. Since $A_1$ is a scalar on each
of these blocks, this can be done without 
affecting the fact that $A_1$ is diagonal.
This process can be continued until all the matrices have been
diagonalized.)
\bigskip
\item[(8)] Every matrix $A$ is a sum $A=B+N$ where
$B$ is diagonalizable, $N$ is ``nilpotent'',\footnote{%
$N^k=0$ for some $k$.} 
and
$BN=NB$. (Reason: By changing basis, we may assume that $A$
is in Jordan form. Then let $B$ be the diagonal part of $A$,
and let $N$ be the strictly upper triangular
part of $A$ containing only $0$'s and $1$'s.)
\end{enumerate}

\vfill

\end{document}

