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\section{Matrix Multiplication 
}

\vspace{1pt}

\subsection{Definitions}

\begin{itemize}
\item Definition MVP Matrix-Vector Product: Suppose $A$ is an $m\times n$
matrix with columns $A_{1},A_{2},A_{3},...,A_{n}$ and $\vec{u}$ is a vector of
size $n$. Then the matrix-vector product of $A$ with $\vec{u}$ is the linear
combination $A\vec{u}=[u]_{1}A_{1}+[u]_{2}A_{2}+[u]_{3}A_{3}+ \cdots
+[u]_{n}A_{n}$

\item Definition MM Matrix Multiplication: Suppose $A$ is an $m\times n$
matrix and $B$ is an $n\times p$ matrix with columns $%
\vec{B}_{1},\vec{B}_{2},\vec{B}_{3},...,\vec{B}_{p}$. Then the matrix product of $A$ with $B$ is the 
$m\times p$ matrix where column $i$ is the matrix-vector product $A\vec{B}_{i}$.
Symbolically, $%
AB=A[\vec{B}_{1}|\vec{B}_{2}|\vec{B}_{3}|...|\vec{B}_{p}]=[A\vec{B}_{1}|A\vec{B}_{2}|A\vec{B}_{3}|...|A\vec{B}_{p}]$ .

\item Definition
\end{itemize}

\subsection{Notation}

\subsection{Proof Techniques}

\subsection{Theorems (Corollaries, Lemmas, Propositions)}

\begin{theorem}
Theorem SLEMM Systems of Linear Equations as Matrix Multiplication:\
Solutions to the linear system $LS(A,\vec{b})$ are the solutions for $\vec{x}$ in the
vector equation $A\vec{x}=\vec{b}$.
\end{theorem}

\begin{theorem}
Theorem EMMVP Equal Matrices and Matrix-Vector Products:\ Suppose that $A$
and $B$ are $m\times n$ matrices such that $Ax=Bx$ for every $x\in C^{n}$.
Then $A=B$.

\begin{enumerate}
\item \textbf{Introduce Block Multiplication here}
\end{enumerate}
\end{theorem}

\begin{theorem}
Theorem EMP Entries of Matrix Products:\ Suppose $A$ is an $m\times n$
matrix and $B$ is an $n\times p$ matrix. Then for $1 \leq i \leq m,1 \leq j \leq
p$, the individual entries of $AB$ are given by $%
[AB]_{ij}=[A]_{i1}[B]_{1j}+[A]_{i2}[B]_{2j}+[A]_{i3}[B]_{3j}+ \cdots
[A]_{in}[B]_{nj}=\sum_{i=1}^{n}[A]_{ik}[B]_{kj}$.
\end{theorem}

\begin{theorem}
Theorem MMZM Matrix Multiplication and the Zero Matrix: Suppose $A$ is an $%
m\times n$ matrix. Then

\begin{enumerate}
\item $AO_{n\times p}=O_{m\times p}$

\item $O_{p\times m}A=O_{p\times n}$
\end{enumerate}
\end{theorem}

\begin{theorem}
Theorem MMIM Matrix Multiplication and Identity Matrix: Suppose $A$ is an $%
m\times n$ matrix. Then

\begin{enumerate}
\item $AI_{n}=A$

\item $I_{m}A=A$
\end{enumerate}
\end{theorem}

\begin{theorem}
Theorem MMDAA Matrix Multiplication Distributes Across Addition:\ Suppose $A$
is an $m\times n$ matrix and $B$ and $C$ are $n\times p$ matrices and $D$ is
a $p\times s$ matrix. Then

\begin{enumerate}
\item $A(B+C)=AB+AC$

\item $(B+C)D=BD+CD$
\end{enumerate}
\end{theorem}

\begin{theorem}
Theorem MMSMM Matrix Multiplication and Scalar Matrix Multiplication:\
Suppose $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Let $%
\alpha $ be a scalar. Then $\alpha (AB)=(\alpha A)B=A(\alpha B).$
\end{theorem}

\begin{theorem}
Theorem MMA Matrix Multiplication is Associative: Suppose $A$ is an $m\times
n$ matrix, $B$ is an $n\times p$ matrix and $D$ is a $p\times s$ matrix.
Then $A(BD)=(AB)D.$
\end{theorem}

\begin{theorem}
Theorem MMIP Matrix Multiplication and Inner Products:\ If we consider the
vectors $u,v\in C^{m}$ as $m\times 1$ matrices then $<u,v>~=u^{t}\bar{v}$
\end{theorem}

\begin{theorem}
Theorem MMCC Matrix Multiplication and Complex Conjugation:\ Suppose $A$ is
an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Then $AB=AB.$
\end{theorem}

\begin{theorem}
Theorem MMT Matrix Multiplication and Transposes:\ Suppose $A$ is an $%
m\times n$ matrix and $B$ is an $n\times p$ matrix. Then $%
(AB)^{t}=B^{t}A^{t}.$
\end{theorem}

\begin{theorem}
Theorem MMAD:\ Matrix Multiplication and Adjoints:\ Suppose $A$ is an $%
m\times n$ matrix and $B$ is an $n\times p$ matrix. Then $(AB)^{\ast
}=B^{\ast }A^{\ast }.$
\end{theorem}

\begin{theorem}
Theorem AIP:\ Adjoint and Inner Product:Suppose $A$ is an $m\times n$ matrix
and $\vec{x}\in \mathbf{C}^{n},$ $y$ $\in $ $\mathbf{C}^{m}$. Then $%
\left\langle A\vec{x},\vec{y}\right\rangle =\left\langle \vec{x},A^{\ast }%
\vec{y}\right\rangle .$
\end{theorem}

\subsection{Examples}

\begin{itemize}
\item Multiplications two ways

\item Show $Ae_{i}=A_{i}$

\item Latest Project:\ $\left[ 
\begin{array}{cc}
\cos \left( \theta \right) & -\sin \left( \theta \right) \\ 
\sin \left( \theta \right) & \cos \left( \theta \right)%
\end{array}%
\right] \left[ 
\begin{array}{c}
x \\ 
y%
\end{array}%
\right] $

\item \textbf{Block Multiplication!}
\end{itemize}

\subsection{Mentioned}

\end{document}
