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\chapter{Math: Abstract Algebra}\label{ch:group}
This chapter focuses on some of the abstractions in math, especially
as they apply to group theory.
Groups appear often in physics:
Boosts and rotations in special relativity
generate the Lorentz group, sets of gauge transformations equipped with
function composition form gauge groups, etc. Some of this presentation
follows sections of Dummit and Foote~\cite{dummit_abstract_2004} and
Georgi~\cite{georgi_lie_1999}.
In those presentations, the connection between groups and linear
transformations is emphasized; therefore we also include
some reminders of linear algebra.
\section{Preliminaries}
We want to discuss math in a fairly abstract way, in part because one of our
goals will be to look at generic mathematical structures. At the end of the book
we will look at groups, structures that have only three characteristics. We will
see that many math structures you've already encountered are examples of
groups
Hence we will need some notation that is agnostic to any particular math
structure.
To start, a {\it set}\index{set} is a collection of objects of
any kind, anything\footnote{This is technically wrong. It turns out that one has
to be very careful how a set is defined. Bertrand Russell was the first do
discover this; if you are interested in what can go wrong, look up
Russell's\index{Russell's paradox}
paradox. For our purposes, this subtlety won't matter.}
you can think of. We usually denote a set
with curly brackets
\begin{equation}
A=\{1,2,3,4\}.
\end{equation}
The things that the set contains are called {\it elements}\index{element}
or\index{member} {\it members}, and we indicate that, e.g. 1 is an element of
$A$ by writing
\begin{equation}
1\in A,
\end{equation}
which we read as ``1 is an element/member of the set $A$" or ``1 is in $A$".
If we want to refer to an arbitrary element $a$ of $A$, we write
\begin{equation}
a\in A,
\end{equation}
and if we want to say that $b$ is not a member of $A$, we write
\begin{equation}
b\notin A,
\end{equation}
The number of elements in a finite set is called its\index{cardinality}
{\it cardinality}. The cardinality of the set $A$ above is 4, and we write
\begin{equation}
|A|=4.
\end{equation}
A {\it finite} set has a finite cardinality; otherwise it's an infinite set.
It is also useful to introduce an organizational hierarchy for sets. For
instance the set
\begin{equation}
B=\{1,2,3,4,5,6\}
\end{equation}
is larger than $A$, but it contains $A$ entirely. We write
\begin{equation}
A\subset B.
\end{equation}
Talking about set logic is a good opportunity to introduce some more notation.
It is common to use $\Forall$ as shorthand for ``for all" or ``for any".
Since $A\subset B$, we know that $\Forall a\in A$ we must have $a\in B$.
To express that last idea, we write
\begin{equation}\label{eq:subsetIn}
a\in A\Rightarrow a\in B.
\end{equation}
The\index{set!empty} {\it empty set} $\emptyset$ is the set with nothing in it
at all. If a set has at least one element, it is\index{nonempty} {\it nonempty}.
From \equatref{eq:subsetIn} it follows that for any set $C$
\begin{equation}
\emptyset\subset C~~~~\text{and}~~~~C\subset C.
\end{equation}
A\index{function}
{\it function} or\index{map} {\it mapping} $f$ between $X$ and $Y$
associates $\Forall x\in X$ a unique element $f(x)\in Y$. We
write
\begin{equation}
f:X\to Y.
\end{equation}
Usually $X$ is called the\index{domain} {\it domain}
and $Y$ is called the\index{codomain} {\it codomain}.
The\index{image} {\it image} or\index{range} {\it range}
is $f(X)$, i.e. the set of all values $f$
maps to given the domain. Note $f(X)\subset Y$.
Next we classify a few different kinds of functions.
A function is\index{injection} {\it injective}
or\index{one-to-one} {\it one-to-one} if each element in $X$
maps to a different element in $Y$. We can express this symbolically as
\begin{equation}
\Forall x_1,x_2\in X,~f(x_1)=f(x_2)\Rightarrow x_1=x_2.
\end{equation}
A\index{surjection} {\it surjective} or\index{onto} {\it onto} function
maps to every possible element of the codomain $Y$. Symbolically,
\begin{equation}
\Forall y\in Y,~\Exists x\in X \suchthat y=f(x),
\end{equation}
where we have introduced the symbol $\Exists$ as shorthand for ``there is at
least one" or ``there exists". Finally a function is\index{function!bijective}
{\it bijective} if it is both an injection and a surjection. In this case each
element of $X$ corresponds to exactly one element of $Y$, and vice-versa.
Pictorial representations of these kinds of functions are shown in
\figref{fig:mapping}.
\begin{figure}
\centering
\includegraphics[width=0.45\linewidth]{figs/Injection.png}~~~
\includegraphics[width=0.45\linewidth]{figs/Surjection.png}\\
\includegraphics[width=0.45\linewidth]{figs/Bijection.png}~~~
\includegraphics[width=0.45\linewidth]{figs/Not-Injection-Surjection.png}
\caption{Example mappings that are injective (top left), surjective (top right),
bijective (bottom left), and none of these (bottom right). Images taken from
Wikipedia~\cite{wiki:bijection}.}
\label{fig:mapping}
\end{figure}
To close out the section, we discuss a couple ways of putting sets together.
The most straightforward way is just to ``add" the sets, i.e. form a new set
whose elements include all the elements from the parent sets. For instance if
you have two sets $A$ and $B$, then the\index{union} {\it union} is
\begin{equation}
A\cup B = \{x\suchthat x\in A~\text{or}~x\in B\}.
\end{equation}
You can also create a set that contains only the elements that both $A$ and $B$ have
in common. This set is the\index{intersection} {\it intersection}
\begin{equation}
A\cap B = \{x\suchthat x\in A~\text{and}~x\in B\}.
\end{equation}
We exist in nature in a space of more than one dimension; therefore it is useful to be
able to combine sets into coordinates. We define the\index{Cartesian product}
{\it Cartesian product} of $A$ and $B$ as the set of tuples
\begin{equation}
A\times B=\{(a,b)\suchthat a\in A~\text{and}~b\in B\}.
\end{equation}
\section{Groups}\label{sec:gpprelim}
A {\it binary operation}\index{binary operation} $\bullet$ on a set
$G$ is a function $\bullet : G\times G\to G$. A {\it group}\index{group},
then, is a set $G$ equipped with a binary operation $\bullet$ that satisfies
the following axioms:
\begin{enumerate}
\item $\bullet$ is associative.
\item $\Exists \id \in G$ such that $\Forall g \in G$,
\begin{equation}
\id\bullet g=g\bullet \id=g.
\end{equation} This element $\id$ is called the {\it identity}.
\index{identity}
\item $\Forall g \in G\ \ \Exists g^{-1} \in G$, called the
{\it inverse}\index{inverse} of $g$, such that
\begin{equation}
g^{-1}\bullet g=g\bullet g^{-1}=\id.
\end{equation}
\end{enumerate}
If group elements commute under $\bullet$ the group is said to be
{\it abelian}\index{abelian}. The {\it order}\index{order} of a group,
denoted $|G|$, is the number of unique elements in the group\footnote{This
is at least true for finite groups.}. A {\it subgroup}\index{subgroup}
$H$ of $G$ is a non-empty subset of $G$ that itself forms a group under
$\bullet$ and in this case we will write $H\leq G$.
(It should be clear from context whether this symbol indicates group
organization or magnitude.) Finally a group is {\it cyclic}\index{cyclic}
if it is generated by a single element; that is, if $\Exists g\in G$ such that
$G=\{g^{n}\suchthat n\in\Z\}$.
It's actually not too common for mathematicians or physicists to write the
$\bullet$ explicitly when showing the composition of two elements. So for
example you will often see $gh$ as shorthand for $g\bullet h$. In general I will
only refer to operations on algebraic structures explicitly when giving the
definition of that structure. Therefore you can expect to see $gh$ instead of
$g\bullet h$ from here on out.
\begin{proposition}{}{}
A subset H of G is a subgroup of G if and only if
$$a,b\in H\Rightarrow ab^{-1}\in H$$
\begin{proof}
($\Rightarrow$) Follows immediately from the definition of a subgroup. To
show ($\Leftarrow$) let $b\in H$. Then by the above conditional, $bb^{-1}
\in H$, which shows $\id\in H$. To show the existence of inverses in $H$,
note $\id,b\in H\Rightarrow \id b^{-1}\in H\Rightarrow b^{-1}\in H$.
Finally, associativity is inherited from G.
\end{proof}
\end{proposition}
An {\it equivalence relation}\index{equivalence relation} $\sim$ on a
set $G$ is a binary operation that has the following properties
$\Forall x,y,z\in G$:
\begin{enumerate}
\item it is {\it reflexive}\index{reflexive}, i.e. $x\sim x$;
\item it is {\it symmetric}\index{symmetric}, i.e.
$x\sim y\Leftrightarrow y\sim x$; and
\item it is {\it transitive}\index{transitive}, i.e. $x\sim y$ and
$y\sim z$ $\Rightarrow$ $x\sim z$.
\end{enumerate}
Let $g\in G$. The set
\begin{equation}
\bar{g}\equiv\{x\in G\suchthat x\sim g\}
\end{equation}
is called an\index{equivalence class}
{\it equivalence class}.
\begin{example*}{}{}
\begin{enumerate}
\item $\Z,\ \Q,\ \R$, and $\C$ are all groups
under addition, and
\begin{equation}
\Z \leq \Q \leq \R \leq \C.
\end{equation}
Each of these sets with 0 removed forms a group under
multiplication. (We have to remove 0 because it has no multiplicative
inverse.)
\item Let $n\in \N$ and define an equivalence relation on
$\Z$ by
\begin{equation}
a\equiv b\ (mod\ n)\ \Leftrightarrow\ n\ |\ (b-a).
\end{equation}
(We read this as ``$a$ is congruent to $b$ modulo $ n$" or ``$a$ is
congruent to $b$ mod $n$.") Define an equivalence class by
\begin{equation}
\bar{a}\coloneqq\{a+mn \suchthat m\in \Z\}.
\end{equation}
The set of all such equivalence classes is called {\it the integers
modulo n}\index{modular arithmetic} and is denoted by
$\Z/n\Z$ or $\Z_n$. It forms a group
under the ``addition" operation exemplified below. As a concrete
illustration, take $\Z_4$. It is of order 4, with
elements $\bar{0},\ \bar{1},\ \bar{2}$, and $\bar{3}$. To see how
addition works, note
\begin{equation}
\label{eq:chgtmaad1}
\begin{aligned}
\bar{1}+\bar{2}&=\{(1+2)+4(m_{1}+m_{2}) \suchthat m_{1},m_{2}
\in \Z\}\\
&=\{3+4m \suchthat m\in \Z\}\\
&=\bar{3}
\end{aligned}
\end{equation}
and
\begin{equation}
\label{eq:chgtmaad2}
\begin{aligned}
\bar{3}+\bar{3}&=\{(3+3)+4(m_{3}+m_{4})\suchthat m_{3},m_{4}
\in \Z\}\\
&=\{2+4(1+m_{3}+m_{4})\suchthat m_{3},m_{4}
\in \Z\}\\
&=\{2+4m\suchthat m\in \Z\}\\
&=\bar{2}
\end{aligned}
\end{equation}
It should be clear from the above that $\bar{0}$ is the identity
element and $\Z_n$ is abelian and cyclic. I would
also like to emphasize that the addition defined in
\equatref{eq:chgtmaad1} and \eqref{eq:chgtmaad2} is not the same
addition as over the integers, even though I have chosen the same
symbol for both cases. One should always be careful of what group
operation is meant when the author is being lazy.
\item Sets of objects besides numbers also form groups. For example let $V$
be any non-empty set of objects and let $S_{V}$ be the set of all
permutations of $V$. Then $S_{V}$ forms a group under function
composition called the {\it symmetric group}\index{group!symmetric}
or {\it permutation group}\index{group!permutation} on the set $V$.
The symmetric group on the $n$ integers $\{1,2,...,n\}$ will be
denoted here as $S_n$. Its order is $n!$.
\end{enumerate}
\end{example*}
In many situations, one encounters two groups of the same order that behave in
essentially the same ways. In a sense, one group is the same the original group,
masquerading about as a unique mathematical object. We could recover the
original group by merely relabeling elements of the masked group and viewing
its operation differently. Let us make these ideas more precise.
Let $G$ and $H$ be groups with operations $\bullet$ and
$\circ$, respectively. A map $\phi : G\to H$ satisfying
\begin{enumerate}
\item $\phi(\id_G)=\id_H$ and
\item $\phi(g \bullet h)=\phi(g)\circ\phi(h)$
\end{enumerate}
$\Forall g,h\in G$ is called a {\it homomorphism}.\index{homomorphism}
If in addition to the above property we know that $\phi$ is a bijection,
$\phi$ is said to be an {\it isomorphism}\index{isomorphism} of $G$ and
$H$, we say that $G$ and $H$ are {\it isomorphic}, and we write $G\cong H$.
Finally an {\it endomorphism}\index{endomorphism} is a homomorphism
mapping a group to itself, and an {\it automorphism}\index{automorphism}
is a bijective endomorphism (isomorphism from a group to itself).
The above definition shows us that when $G\cong H$, $H$ is really just $G$ in
disguise. Since $\phi$ is a bijection we have a way of associating each element
of $H$ with exactly one element of $G$ and vice-versa (one could consider
$\phi$ the relabeling), and the fact that $\phi$ is a homomorphism shows us
that the binary operations of the groups act the same way.
We note that if $G$ and $H$ are not isomorphic, it can be that $\phi$ maps
multiple elements of $G$ to the same element in $H$. The {\it
kernel}\index{kernel} of $\phi$, $\ker\phi$, is the set of elements in $G$ that map
to $\id_H$.
A {\it field}\footnote{There are many instances where a term used in
a mathematical context is slightly different than its meaning in
a physics context. In a physics context, ``field" means something like
``a math object whose domain is every space-time point".
Due to these differing conventions, one usually has to determine based
on context which kind of field is meant.}\index{field}
is a set $F$ equipped with two binary
operations $+$ and $\cdot$ satisfying the following properties:
\begin{enumerate}
\item $F$ equipped with $+$ is an abelian group. (Denote the
identity element with respect to this operation 0. 0 is often
referred to as the\index{identity!additive} {\it additive identity}.)
\item $F-\{0\}$ equipped with $\cdot$ is an abelian group. (Denote the
identity element with respect to this operation 1. As you may have
guessed, 1 is called the\index{identity!multiplicative}
{\it multiplicative identity}.)
\item $a\cdot (b+c)=(a\cdot b)+(a\cdot c)\ \ \Forall a,b,c\in F$. This is
called the\index{distributive property} {\it distributive property}.
\end{enumerate}
A {\it subfield}\index{subfield}
is a subset $A$ of $F$ whose elements still form a field.
Later we will see that fields are the fundamental sets out of
which vectors are created.
\begin{example*}{}{}
\begin{enumerate}
\item For any group $G$, $G\cong G$.
\item The rotational symmetries of a regular $n$-gon are isomorphic to
$\Z_n$.
\item $\Q$ is a subfield of $\R$, which is in turn a subfield of $\C$.
\end{enumerate}
\end{example*}
Let $G_1$, $G_2$, ... $G_n$ be groups. The
{\it direct product}\index{direct!product}\index{product!direct}
$G_1\times G_2\times ...\times G_n$
is the set of tuples $(g_1,g_2,...,g_n)$ with $g_i\in G_i$
and the group operation defined component-wise like
\begin{equation}
(g_1,g_2,...,g_n)(h_1,h_2,...,h_n)
=(g_1h_1, g_2h_2,...,g_nh_n),
\end{equation}
where $h_i\in G_i$ also.
In this definition, the $i\nth$ component of the product
uses the group operation of the $i\nth$ group. Composing
a new group from other groups in this way ensures that
the direct product is also a group, as it inherits its
groupness from them. Hence it is a Cartesian product
that preserves group structure.
\begin{proposition}{}{}
$G_1\times G_2\times ...\times G_n$ is a group.
If all $G_i$ are finite, its order is $\Pi_i |G_i|$.
\end{proposition}
Additionally it's clear that each $G_i$ can be considered
a subgroup of the direct product. For instance $G_1$ is
isomorphic to the group of tuples
\begin{equation}
(g_1,\id,...,\id).
\end{equation}
It's also useful to note that the direct product is
isomorphic up to any permutation of its factors.
This notion of a direct product can be extended also
up to infinitely many groups. Still the group operation
is defined component wise.
The {\it direct sum}\index{direct!sum} is a subgroup
of the direct product. It is the restriction to elements of
the direct product that are $\id$ in all but finitely
many of the components. For finitely many abelian groups,
the direct sum and direct product are the same. The
salient feature of a direct sum is that it introduces
some notion of independence: Enforcing that component $i$
is $\id$ enforces that this group element of the
direct sum ``does not speak" to group $G_i$.
\section{Quotient groups}\label{sec:q}\index{quotient group}
In this section we follow Ref.~\cite{dummit_abstract_2004}, which gives a nice
presentation of quotient groups. There, they point out that understanding
quotient groups more or less amounts to understanding group homomorphisms.
To begin we consider two groups $G$ and $H$ with a homomorphism $\phi:G\to H$.
The {\it fibers}\index{fiber} of $\phi$ are the sets of elements in $G$ mapping
to a single element in $H$, which is depicted graphically in
\figref{fig:quotientGroup}. If $a$ represents one of the dots in $H$,
the solid line above $a$ is the ``fiber of $\phi$ over $a$", which we can represent
at $\phi^{-1}(a)$.
Since $\phi$ is a homomorphism, it follows that however multiplication works
among the dots in $H$, multiplication among the fibers must function the same
way. This suggests the creation of a new group, the {\it quotient group},
formed from the fibers. By construction, the quotient group with this
multiplication is clearly isomorphic to the image of $G$ under $\phi$.
\begin{figure}
\centering
\includegraphics[width=0.60\linewidth]{figs/quotientGroup.pdf}
\caption{Graphical depiction of constructing a quotient group from a group
$G$. The fibers, which are the solid, vertical lines, are mapped by the homomorphism
$\phi$ into $H$. The set of fibers forms the quotient group. Image taken
from Ref.~\cite{dummit_abstract_2004}.}
\label{fig:quotientGroup}
\end{figure}
Let $H\le G$ and $a\in G$. The {\it left coset}\index{coset} of $H$ with respect to
$a$ is the set
\begin{equation}
aH=\{ah\suchthat h\in H\}.
\end{equation}
The {\it right coset} $Ha$ is defined similarly, but with $h$ and $a$
interchanged. You can think of e.g. a left coset of $H$ as a left translate of
$H$ by $a$, and similarly for the right coset. Any element of a coset is
called a {\it representative} of the coset.
We are now going to see that cosets define equivalence classes. To begin with
let $H\le G$; $x,y\in H$; and define a binary relation\footnote{You can
heuristically think of this equation as the statement that if I go ``forward" by
$x$, then ``backward" by $y$, I remain in $H$.} $\sim$ by
\begin{equation}\label{eq:cosetEquivalence}
x\sim y \Leftrightarrow xy^{-1}\in H.
\end{equation}
\begin{proposition}{}{}
$\sim$ is an equivalence relation.
\begin{proof}
We just need to check that it satisfies all the defining properties.
\begin{enumerate}
\item $xx^{-1}=\id$ $\Rightarrow$ $x\sim x$.
\item $x\sim y$ $\Rightarrow$ $xy^{-1}\in H$ $\Rightarrow$
$(xy^{-1})^{-1}\in H$ $\Rightarrow$ $yx^{-1}\in H$
since $H$ is a group, so its elements have inverses.
Hence $y\sim x$.
\item $x\sim y$ and $y\sim z$ $\Rightarrow$ $xy^{-1}\in H$ and
$yz^{-1}\in H$. Then since $H$ is closed under multiplication,
$xy^{-1}yz^{-1}=xz^{-1}\in H$ $\Rightarrow$ $x\sim z$.
\end{enumerate}
\end{proof}
\end{proposition}
According to definition~\eqref{eq:cosetEquivalence}, given $H$,
the equivalence classes of $G$ under $\sim$ are the sets
$A\subset G$ satisfying $xy^{-1}\in H$ $\Forall x,y\in A$. To see why this is
important, fix $a\in A$. Then $\Forall x\in A$,
\begin{equation}
\begin{aligned}
x\sim a &\Rightarrow xa^{-1}\in H \\
&\Rightarrow xa^{-1}=h~\text{for some}~h\in H\\
&\Rightarrow x=ha\\
&\Rightarrow A\subset Ha.
\end{aligned}
\end{equation}
In other words, the equivalence classes are the right cosets of $H$! If we
instead defined the equivalence relation by $x\sim y$ $\Leftrightarrow$
$x^{-1}y\in H$, we would have found the equivalence classes to be the left
cosets of $H$.
\begin{proposition}{}{cosetPartition}
The cosets of $H$ partition $G$.
\begin{proof}
Let $g\in G$. We have to check that (1) $g$ belongs to some coset of $H$,
and further that (2) $g$ does not belong to two different cosets.
\begin{enumerate}
\item Let $h\in H$. Then $g=h(h^{-1}g)$, i.e. it's in the right
coset of $H$. A similar computation shows it's in the left
coset of $H$ as well.
\item Let $x,y\in G$. If $g\sim x$ and $g\sim y$, i.e. it belongs in
the $x$-coset and $y$-coset of $H$, then $x\sim y$ since
$\sim$ is transitive.
\end{enumerate}
\end{proof}
\end{proposition}
We can now construct our new group. The group is a collection of cosets, but
it only forms a group if the cosets are special. In particular,
a subgroup $N\le G$ is said to be {\it normal}\index{normal} if
$\Forall g\in G,$ $gN=Ng$. In this case we write $N\unlhd G$. The
{\it quotient group}\index{quotient group} $G/N$ is the set of all cosets of
$N$. We define an operation on $G/N$ by
\begin{equation}
(xN)(yN)=x(Ny)N=x(yN)N=(xy)(NN)=(xy)N.
\end{equation}
Hence we see why it was so crucial that the subgroup be normal: It allowed
us to move the $y$ past $N$ in the second step, thus guaranteeing $G/N$ is
closed under this operation.
Moreover by \propref{prp:cosetPartition}, $G/N$ partitions $G$.
\begin{proposition}{}{}
$G/N$ forms a group under the above operation.
\begin{proof}
This is pretty clearly a group because $G$ is. The identity is $\id N$,
and each $xN$ has inverse $x^{-1}N$
\end{proof}
\end{proposition}
To summarize: If we find a normal subgroup $N$ of $G$, we can make a new group
$G/N$ by partitioning $G$ into disjoint cosets of $N$, which turn out to be
disjoint equivalence classes. These quotient groups are useful and pop up
pretty frequently; indeed the group $\Z/n\Z$, which we
encountered in the first section, is a quotient group. Now you understand the
notation.
\begin{example*}{}{}
\index{modular arithmetic}
Take $G=(\Z,+)$ and $N=4\Z$. Clearly $4\Z$ is a
normal subgroup of $\Z$ since it's commutative. To form the
quotient group $\Z_4=\Z/4\Z$, we find all cosets of $4\Z$.
These are
\begin{itemize}
\item $4\Z$,
\item $4\Z+1$=$\{4m+1\suchthat m\in\Z\}$,
\item $4\Z+2$, and
\item $4\Z+3$.
\end{itemize}
To check for instance that $4\Z+1$ is an equivalence class, we just
need to show that $x,y\in4\Z+1\Rightarrow xy^{-1}\in4\Z$.
Let $x=4m+1$ and $y=4n+1$ for some $m,n\in\Z$. Then
$$xy^{-1}=x-y=4m+1-(4n+1)=4(m-n)\in4\Z.$$
(This is an instance where the notation can be confusing; remember
$\Z$ is being considered as a group under addition, so applying the
inverse group operation means subtracting.) To see that there are no other
cosets, note that $4\Z+4$=$4\Z$.
\end{example*}
To close out this Section, we prove Lagrange's theorem, which often useful
for learning about group structure.
\begin{theorem}{Lagrange}{lagrange}\index{Lagrange's theorem}
Let $G$ be a finite group and let $H\leq G$. Then $|H|$ divides $|G|$.
In particular, the number of cosets of $H$ in $G$ is $|G|/|H|$.
\begin{proof}
Note that $|H|=|gH|$ for any $g\in G$, i.e. all cosets have the same
order. On the other hand, by \propref{prp:cosetPartition}, the cosets
partition $G$. Hence $|G|=n|H|$, where $n$ is the number of cosets of $H$.
\end{proof}
\end{theorem}
\section{Vectors and vector spaces}\label{sec:vectors}
This section begins a brief foray into linear algebra.
Besides the fact that linear algebra has many applications to
physics on its own, it is useful for understanding group
representations, which we will discuss in \secref{sec:represent}.
Let $F$ be a field. A {\it vector space}\index{vector!space} over $F$ is a set
$V$ together with a binary operation $+$ under which $F$ is abelian and a
mapping $F\times V\to V$, denoted $av\ \Forall a\in F$ and $v\in V$,
satisfying
\begin{enumerate}
\item $(a+b)v=av+bv$,
\item $(ab)v=a(bv)$,
\item $a(v+w)=av+aw$, and
\item $1v=v$
\end{enumerate}
$\Forall a,b\in F$ and $v,w\in V$. The elements $v$ and $w$ are\index{vector}
called {\it vectors}\footnote{Note that within the contexts of high
energy physics and relativity, the word ``vector" has a slightly more
specific meaning: A ``vector" in those contexts is a vector in the
mathematical sense along with a demand on how it behaves under
Lorentz transformations.} A {\it subspace}\index{subspace}
is a subset $W$ of
$V$ that still forms a vector space over $F$. I will become immediately
less formal and refer to the underlying field only if it's absolutely
necessary.
Let $V$ and $V'$ be vector spaces over a field $F$. A
{\it linear transformation}\index{transformation!linear}
of $V$ to $V'$ is a mapping $T : V\to V'$ such that
\begin{enumerate}
\item $T(v+w)=T(v)+T(w)$ and
\item $T(av)=aT(v)$
\end{enumerate}
$\Forall a\in F$ and $v,w\in V$. We will often drop the parentheses
and simply write $T(v)=Tv$. A linear transformation mapping a
vector space to itself is called a {\it linear operator}.
\index{operator!linear}\index{null space}\index{kernel}
The {\it null space} or {\it kernel} of $T$ is
\begin{equation}
\mathcal{N}(T)\equiv\{v\in V:Tv=0\};
\end{equation}
i.e. it is the set of vectors annihilated by $T$. Finally $T$ is said
to be {\it idempotent}\index{idempotent} if $T^2=T\neq0$.
To connect some of the above definitions to what we learned in
\chref{ch:group}, we see that a linear operator is an endomorphism.
Furthermore we see that if a linear operator has an inverse, then it
is an automorphism.
\index{function!distance}\index{metric}
Let $M$ be some set. A {\it distance function} or {\it metric}
is a function
$d:M\times M\to \R$
such that $\Forall x,y,z\in M$
\begin{enumerate}
\item $d(x,y)=0\Leftrightarrow x=y$,
\item $d(x,y)=d(y,x)$,
\index{triangle inequality}
\item $d(x,z)\leq d(x,y)+d(y,z)$ (this is called the {\it triangle
inequality}).
\end{enumerate}
\index{metric space}
A {\it metric space}, then, is a set $M$ equipped with a metric $d$.
One can generalize the idea of a metric and relax some of these
conditions. I think for most purposes in physics this definition is sufficient.
\begin{example*}{}{}
$\R^n$ forms a metric space when equipped with
\begin{equation}
d(x,y)=|x-y|=\sqrt{\sum\limits_{i=1}^n(x_i-y_i)^2}.
\end{equation}
\index{metric!Euclidean}
This known as the {\it Euclidean metric}.
\end{example*}
Next we turn to the concepts of orthogonality and dimensionality, which
essentially tell us when two vectors are independent and how many independent
generators it takes to make a vector space.
We will formulate these ideas in terms of inner products.
\index{product!inner}\index{product!scalar}
Let $V$ be a vector space over $\C$. An {\it inner
product} or {\it scalar product} is a mapping
$(\ ,\ ):V\times V\to \C$
satisfying $\Forall x,y,z\in V$ and $\alpha \in F$
\begin{enumerate}
\item $(x,y)=(y,x)^{*}$,
\item $(x,y+z)=(x,y)+(x,z)$
\item $(x,\alpha y)=\alpha(x,y)$
\end{enumerate}
Note that the above properties also imply $(x+y,z)=(x,z)+(y,z)$ and
\index{vector!magnitude}\index{vector!norm}
$(\alpha x,y)=\alpha^{*}(x,y)$. The {\it magnitude} or {\it norm}
of a vector $x$ is $|x|\equiv(x,x)^{1/2}$. If $(x,y)=0$ then $x$
\index{orthogonal}\index{orthonormal}
and $y$ are said to be {\it orthogonal} and we write $x\perp y$.
If in addition $|x|=|y|=1$, then they are {\it orthonormal}.
\begin{proposition}{}{orthog}
Let $V$ be a vector space with inner product and let $W$ a subspace
of $V$ with basis $\{w_1,...,w_N\}$. Consider the mapping $P:V\to W$
defined by
$$
Pv=\sum_{i=1}^N\frac{(v,w_i)}{|w_i|^2}w_i.
$$
(This map is the {\it orthogonal
projection}\index{orthogonal!projection}.)
Then $v-Pv\perp W.$
\begin{proof}
We have
\begin{equation*}
\begin{aligned}
(v-Pv,w_j)&=(v,w_j)-(Pv,w_j)\\
&=(v,w_j)-\sum_{i=1}^N\frac{(v,w_i)}{|w_i|^2}(w_i,w_j)\\
&=(v,w_j)-\frac{(v,w_j)}{|w_j|^2}|w_j|^2\\
&=(v,w_j)-(v,w_j)\\
&=0.
\end{aligned}
\end{equation*}
Since $v-Pv$ is orthogonal to every basis vector of $W$, it is
orthogonal to every vector in $W$.
\end{proof}
\end{proposition}
The above proposition gives an algorithm to construct
a vector which is orthogonal to all elements in some subspace.
The algebra was a bit tedious but the intuition is clear:
if I project $v$ onto every basis vector of $W$ and add all
those contributions together, I get the part of $v$ that lies
within $W$. So I just subtract that away.
As with groups, one can build up a direct sum\index{direct!sum}
of vector spaces. For instance with two vector spaces $V$ and $W$,
$V\oplus W$ is the set of vectors $(v,w)$ with $v\in V$ and $w\in W$.
Addition and scalar multiplication are defined like
\begin{equation}\begin{aligned}
(v_1,w_1)+(v_2,w_2)&=(v_1+v_2,w_1+w_2)\\
\alpha(v,w)&=(\alpha v,\alpha w),
\end{aligned}\end{equation}
where $\alpha\in F$. This ensures $V\oplus W$ is a vector space
over $F$ as well.
\section{Matrices}
Matrices play a fundamental role in physics as well.
We will see they are very useful for understanding different kinds of
symmetries, but they have many other applications.
Given a vector space $A^n$, we associate
$n\times n$ matrices\footnote{Such a matrix is usually called a {\it square
matrix}. Matrices do not have to be square matrices, i.e. they can have a
different number of rows than columns. In physics square
matrices matter the most, so that's all we will worry about for these notes.},
which have the form
\begin{equation}\label{eq:basicMatrix}
M=\left(\begin{array}{cccc}
a_{11} & a_{12} & & a_{1n}\\
a_{21} & a_{22} & & a_{2n}\\
& & \ddots& \\
a_{n1} & a_{n2} & & a_{nn}
\end{array}\right).
\end{equation}
We call each entry $a_{ij}\in A$, $1\leq i,j\leq n$, an {\it element}. The position
of each element is indicated by two subscripts, the left
indicating the row number and the right indicating the column number.
Matrices as defined above\footnote{That is, square matrices. By the way,
in this context of mapping vector spaces into themselves, we sometimes
call matrices {\it operators}\index{operator}.} map $A^n$ into itself, i.e.
\begin{equation}\label{eq:operatorDef}
M:A^n\to A^n.
\end{equation}
In general, this mapping is achieved through matrix multiplication. If one
multiplies a vector $v$ with a matrix $M$, one gets for the $i\nth$ component
of the product $Mv$
\begin{equation}\label{eq:matTimesVec}
(Mv)_i=\sum_{j=1}^n a_{ij}v_j,
\end{equation}
where $M$ is our matrix and $v$ is a vector.
Matrices can be added together element-wise.
Analogously as with many kinds of number systems, it is useful to introduce
in $n$ dimensions an additive identity or {\it zero matrix}\index{matrix!zero} as
\begin{equation}
\zd_n=\left(\begin{array}{cccc}
0 & 0 & & 0\\
0 & 0 & & 0\\
& & \ddots& \\
0 & & & 0
\end{array}\right),
\end{equation}
i.e. it's the $n\times n$ matrix with 0 in every element. With our definition
of matrix addition, we find that for any $n\times n$ matrix $M$
\begin{equation}
\zd_n+M=M+\zd_n=M.
\end{equation}
Besides addition, it is useful to be able to carry out multiplication.
The simplest kind is scalar multiplication\index{multiplication!scalar},
which is defined similarly as with vectors.
One can also multiply a matrix $M$ and a matrix $L$. If $L$ has elements
$b_{ij}$, then the product $ML$ is defined element-wise by
\begin{equation}
(ML)_{ij}=\sum_{k=1}^n a_{ik}b_{kj}.
\end{equation}
Notably, matrix multiplication is not in general commutative.
Again to continue our analogy with other number systems,
we introduce a multiplicative identity for matrices.
In $n$ dimensions, the {\it identity matrix} is
\begin{equation}\index{identity!matrix}\index{matrix!identity}
\id_n=\left(\begin{array}{cccc}
1 & 0 & & 0\\
0 & 1 & & 0\\
0 & & \ddots& \\
0 & & & 1
\end{array}\right),
\end{equation}
i.e. it is the matrix with 1 along the diagonal and 0 everywhere else. Using
matrix multiplication you can show that for any $n\times n$ matrix $M$
\begin{equation}
M\id_n =\id_n M = M.
\end{equation}
Sometimes a matrix is {\it invertible}\index{invertible},
i.e. $\Exists M^{-1}$ so that
\begin{equation}
M^{-1} M=\id_n.
\end{equation}
If a matrix is not invertible, it is said to be
{\it singular}\index{singular}.
To round out this discussion of thinking about matrices as math objects in their
own right, it is sometimes useful to define functions of matrices.
For starters, we can always
raise a matrix to an arbitrary power $k\in\N$. Hence we have well defined
functions of the form
\begin{equation}
f(M)=M^k\equiv\underbrace{M\,M\,...\,M}_{k\text{ times}},
\end{equation}
and as with ordinary numbers, we define $M^0\equiv \id_n$.
Using scalar multiplication and matrix addition, we are therefore also
able to sensibly define polynomials of matrices
\begin{equation}
f(M)=\alpha_0\id_n + \alpha_1 M + \alpha_2 M^2...+\alpha_n M^n.
\end{equation}
The fact that we can construct polynomials out of matrices empowers us
to define even more general functions through their Taylor series.
For example, the exponential $\exp:\R\to\R$ is given through its Taylor expansion
as
\begin{equation}
\exp(x)=\sum_{i=0}^\infty \frac{x^k}{k!}.
\end{equation}
This allows us to define the exponential of a matrix as
\begin{equation}\label{eq:expMat}
\exp(M)=\sum_{i=0}^\infty \frac{M^k}{k!}.
\end{equation}
All the typical elementary functions $\sin$, $\sinh$, and so on can be
analogously defined on matrices.
I am obligated to introduce some notation one frequently encounters
in physics with regard to matrices. The {\it trace}\index{trace} of an
$n\times n$ matrix is the sum of its diagonal elements,
\begin{equation}
\tr M=\sum_i^nM_{ii}.
\end{equation}
When you {\it transpose}\index{transpose} a matrix, $M^t$, you interchange all of its
off-diagonal elements; i.e. the $i,j$-element gets replaced with the
$j,i$-element, and vice-versa.
The {\it complex conjugate}\index{complex conjugate}
of a matrix $M^*$ simply conjugates all its elements.
Finally we will sometimes need the conjugate-transpose or
{\it adjoint}\index{adjoint} of a matrix, which is indicated with a
little dagger\footnote{Hence we sometimes say ``$M$-dagger".},
$M^\dagger$. It is
\begin{equation}
M^\dagger\equiv (M^*)^t.
\end{equation}
The {\it determinant}\index{determinant}
of $M$ is
\begin{equation}
\det M\equiv\epsilon_{i_1...i_n}M_{1\,i_1}...M_{n\,i_n},
\end{equation}
where $\epsilon_{i_1...i_n}$ is the Levi-Civita symbol.
An important and often-used fact relating determinants
and inverses is the following:
\begin{proposition}{}{detSingular}
A matrix is invertible if and only if its determinant is nonzero.
\end{proposition}
Matrices will appear in many contexts for us, one of which
is quantum physics. Quantum physics was historically built up partially in
analogy to classical physics. As we were developing a consistent theory for these
small systems, we noticed that the kinds of relations we needed our observables
to obey could not be satisfied by familiar scalar fields like $\R$ and $\C$.
It was realized that we needed non-commutative symbols to understand,
for example, the famous Stern-Gerlach experiment. We already noted that matrix
multiplication is not commutative, so from that perspective matrices make good
candidates. More deeply, groups can be mapped into the set of
vector space automorphisms,
i.e. group elements can be
represented\footnote{See \secref{sec:represent}.}\index{representation} by matrices,
and so we can always think of non-commutative symbols in terms of matrices,
whenever that helps us.
In the context of quantum physics, observables like position are represented by
matrices. But what we measure for an observable cannot depend on how we
represent\footnote{I mean ``represent" here in the colloquial sense.} it,
so we would like information about a matrix that is e.g. independent of
basis choice. The most important example of such information is as follows:
Let $V$ be a vector space $V$ over a field $F$ and let $v\in V$,
and suppose the matrix $M$ satisfies
\begin{equation}\label{eq:eigen}
Mv = \lambda v
\end{equation}
for some $\lambda\in F$. Then $v$ is an {\it eigenvector}\index{eigenvector}
of $M$ with {\it eigenvalue}\index{eigenvalue} $\lambda$.
\begin{proposition}{}{eigenvalue}
A matrix's eigenvalues do not depend on the basis.
\end{proposition}
\propref{prp:eigenvalue} shows that eigenvalues are a great candidate
for measurable quantities, and indeed this is what we use in practice.
Finding eigenvectors and eigenvalues is pretty straightforward.
All eigenvalues of $M$ satisfy the
{\it characteristic polynomial}\index{characteristic polynomial}
\begin{equation}\label{eq:characteristicPolynomial}
\det\left(\lambda\id_n-M\right)=0,
\end{equation}
and in practice that's how one usually computes them.
Once you have found all the $\lambda$, you can plug them
into \equatref{eq:eigen} to find their corresponding
eigenvectors. If \equatref{eq:characteristicPolynomial}
has a root of order $m>1$, that eigenvalue is repeated.
Additionally, multiple vectors may have the same eigenvalue.
In case the eigenvalues are real, a matrix is
{\it positive definite}\index{positive definite}
if its eigenvalues are all positive, and
{\it positive semidefinite}\index{positive semidefinite}
if the eigenvalues are all nonnegative.
A matrix $U$ is said to be {\it unitary}\index{unitary} if
\begin{equation}\label{eq:unitary}
U^\dagger U=UU^\dagger=\id_n,
\end{equation}
i.e. unitary matrices are those whose inverses are the same as
their adjoints. Besides the fact that, by definition, these
matrices are easily invertible, they have the nice property
for any vector $v$ that
\begin{equation}
|Uv|=\sqrt{(Uv,Uv)}=\sqrt{(Uv)^\dagger Uv}=\sqrt{v^2}=|v|,
\end{equation}
i.e. they preserve vector lengths. From
\equatref{eq:unitary} we see also they have determinant 1.
A matrix $H$ is {\it hermitian}\index{hermitian}\footnote{Named
after a French mathematician, Charles Hermite. He's known
for a lot of stuff; for instance he was the first to
show that $e$ is transcendental.} if
\begin{equation}
H^\dagger=H.
\end{equation}
Hermitian matrices by definition have real eigenvalues.
This is useful in the context of quantum physics, since physical observables
correspond to matrices, and eigenvalues of those matrices are measurable
quantities, which ought to be real.
Another nice property of hermitian matrices is the following:
\begin{theorem}{}{hermit}
Hermitian matrices are diagonalizable.
\end{theorem}
A matrix $S$ is {\it symmetric}\index{symmetric} if
\begin{equation}
S^t=S,
\end{equation}
i.e. if the elements are symmetric about the diagonal.
Meanwhile a matrix $O$ is {\it orthogonal}\index{orthogonal} if
\begin{equation}
O^tO=\id_n.
\end{equation}
This is
the real counterpart of a unitary matrix.
\begin{proposition}{}{orthogonalMatrixLength}
Orthogonal matrices preserve length; i.e. if $O$ is
a real, orthogonal matrix and $v\in\R^n$, then
$$
|Ov|=|v|.
$$
\end{proposition}
\subsection{General complex matrices}
Many of the matrices we deal with in physics are unitary, hermitian,
or orthogonal, and hence they have relatively nice properties.
Later in the context of lattice QCD, we will deal with more general
complex matrices that are not guaranteed to have these properties.
Therefore we collect here some useful results for complex matrices.
\index{Cayley-Hamilton theorem}
\begin{theorem}{The Cayley-Hamilton theorem}{cayleyHamilton}
Every square, complex matrix\footnote{Actually this theorem is
a bit more general as it applies to commutative rings.
But our matrices will always be over $\C$, so there is not
much point to introduce rings and make the distinction.}
satisfies its own characteristic equation.
\end{theorem}
The Cayley-Hamilton theorem finds use as a method for
efficient computation of matrix inverses. For
example one can show for a $3\times 3$ matrix
\begin{equation}\label{eq:inv3x3CH}
M^3-(\tr M) M^2+\frac{1}{2}\left((\tr M)^2-\tr M^2\right) M
-\det M \id_3=O.
\end{equation}
One can then multiply through by $M^{-1}$, and hence trade
the general problem of matrix inversion with a handful for matrix
multiplications, which may be faster.
Later we will encounter Lie groups whose representations
are traceless, which simplifies inverting such
matrices substantially.
\index{spectral theorem}
\begin{theorem}{Spectral theorem}{spectral}
Let $S$ be an $n\times n$ symmetric, complex matrix with
eigenvalues $\lambda_i$ and eigenvectors $u_i$. Then
We can decompose $S$ as
$$
S=U^t\Lambda U,
$$
where $\Lambda\equiv\diag\left(\lambda_1,...,\lambda_n\right)$ and
$U=\left(u_1,...,u_n\right)$.
\end{theorem}
The spectral theorem can be employed to find a useful
factorization for an arbitrary matrix, which goes by
the name of
{\it singular value decomposition}\index{singular value decomposition} (SVD).
\begin{theorem}{Singular value decomposition}{SVD}
Any square\footnote{There is a statement also for non-square matrices.
But again, that's that not what we're interested in.}
matrix $M$ can be factorized as
$$
M=USV^t,
$$
where $S=\diag\left(s_1,...,s_n\right)$, $s_i\geq 0$,
and $U$ and $V$ are orthogonal matrices.
\end{theorem}
SVD shows us that we can think of the action of any matrix on a real vector
in three steps. More precisely, since $U$ and
$V^t$ are orthogonal, they can be thought of