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\begin{document}

$
A=
\begin{bmatrix}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{bmatrix},
B=
\begin{bmatrix}
B_{11} & B_{12}\\
B_{21} & B_{22}
\end{bmatrix},
C=
\begin{bmatrix}
C_{11} & C_{12}\\
C_{21} & C_{22}
\end{bmatrix}
$

\medskip
$
A \cdot B = C
$

\medskip
$
C_{11} = A_{11} \cdot B_{11} + A_{12} \cdot B_{21}\\
C_{12} = A_{11} \cdot B_{12} + A_{12} \cdot B_{22}\\
C_{21} = A_{21} \cdot B_{11} + A_{22} \cdot B_{21}\\
C_{22} = A_{21} \cdot B_{12} + A_{22} \cdot B_{22}
$

\medskip
\begin{math}
\begin{aligned}
\text{I}   &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\
\text{II}  &= (A_{21} + A_{22}) \cdot B_{11} \\
\text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\
\text{IV}  &= A_{22} \cdot (-B_{11}+B_{21}) \\
\text{V}   &= (A_{11} + A_{12}) \cdot B_{22} \\
\text{VI}  &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12})) \\
\text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
\end{aligned}
\end{math}


\medskip
\begin{math}
\begin{aligned}
C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\
C_{21} &= \text{II} + \text{IV} \\
C_{12} &= \text{III} + \text{V}\\
C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\
\end{aligned}
\end{math}


\medskip
\begin{math}
\begin{aligned}
C_{11} &= \text{II} + \text{IV} \\
C_{11} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) + A_{22} \cdot (-B_{11}+B_{21}) - (A_{11} + A_{12}) \cdot B_{22} + (A_{12} - A_{22}) \cdot (B_{21} + B_{22})C_{21} \\
C_{11} &= A_{11}B_{11} + A_{11}B_{22} + A_{22}B_{11} + A_{22}B_{22} -A_{22}B_{11}+A_{22}B_{21} - A_{11}B_{22} - A_{12}B_{22}+ A_{12}B_{21} + A_{12}B_{22} - A_{22}B_{21} - A_{22}B_{22} \\
C_{11} &= A_{11}B_{11} + A_{12}B_{21}
\end{aligned}
\end{math}

\section{Winograd}

$
x_1 y_1 + x_2 y_2 = (x_1 +y_2)(y_1 + x_2)-x_1 x_2 - y_1 y_2
$

$
x = (x_1, \cdots, x_n), y=(y_1, \cdots, y_n)
$

\[
\xi = \sum_{j=1}^{ \lfloor n/2 \rfloor} x_{2j-1} \cdot x_{2j}
\]

\[
\eta = \sum_{j=1}^{ \lfloor n/2 \rfloor} y_{2j-1} \cdot y_{2j}
\]

\[
\langle x,y \rangle =
\begin{cases}
 \displaystyle  \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta & \text{if  $n$ is even}\\
\displaystyle  \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta + x_n y_n & \text{if  $n$ is odd}
\end{cases}
\]

\end{document}
