91Ó°ÊÓ

Environment multiplication

Allowing the following matrices:

$X=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$ and $Y=\left[\begin{array}{cc}E& F\\ G& H\end{array}\right]$

X and Y vectors are split into 4 size blocks. $\frac{n}{2}×\frac{n}{2}$. This combination between X or Y matrix (z) has the following i , j the elements because X and Y were $n×n$matrices:

_{$Z_{ij}={\underset{}{∑}}_{k=1}{X}_{ik}{Y}_{kj}$} where $1≤i,j≤n$

With each region of both the process of making the product, the specified property may be demonstrated (Z) .

For the sector in which $i,j≤\frac{n}{2}$:

$$\begin{gathered} Z_{ij}={\left(X,Y\right)}_{ij} \\ =\underset{K=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj} \\ =\underset{k=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj}+\underset{K=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj} \\ =\underset{K=1}{\overset{n/2}{∑}}{A}_{ik}{E}_{kj}+\underset{K=1}{\overset{n/2}{∑}}{B}_{ik}{G}_{kj} \\ ={\left[AE+BG\right]}_{ij} \end{gathered}$$

For the sector in which $i≤\frac{n}{2}$and $\frac{n}{2}≤j≤n$:

$$\begin{gathered} Z_{ij}={\left(XY\right)}_{ij}=\underset{K=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj} \\ =\underset{K=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj}+\underset{n_{k=-+1_{{}_2}}}{\overset{n}{∑}}{X}_{ik}{Y}_{kj} \\ =\underset{K=1}{\overset{n/2}{∑}}{A}_{ik}{F}_{kj}+\underset{K=1}{\overset{n/2}{∑}}{B}_{ik}{H}_{kj} \\ ={\left[AF+BH\right]}_{ij} \end{gathered}$$

For the sector in which $\frac{n}{2}≤i≤n$and $j≤\frac{n}{2}$:

$$\begin{gathered} Z_{ij}={\left(X,Y\right)}_{ij} \\ =\underset{K=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj} \\ =\underset{k=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj}+\underset{K=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj} \\ =\underset{K=1}{\overset{n/2}{∑}}{C}_{ik}{E}_{kj}+\underset{K=1}{\overset{n/2}{∑}}{D}_{ik}{G}_{kj} \\ ={\left[CE+DG\right]}_{ij} \end{gathered}$$

For the sector in which $\frac{n}{2}≤i,j≤n$ :

$$\begin{gathered} Z_{ij}={\left(X,Y\right)}_{ij} \\ =\underset{K=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj} \\ =\underset{k=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj}+\underset{K=1}{\overset{n}{∑}}{X}_{ik}{Y}_{kj} \\ =\underset{K=1}{\overset{n/2}{∑}}{C}_{ik}{F}_{kj}+\underset{K=1}{\overset{n/2}{∑}}{D}_{ik}{H}_{kj} \\ ={\left[CF+DH\right]}_{ij} \end{gathered}$$

The product of X and Y can be expressed as follows:

$Z=\left[\begin{array}{cc}{Z}_{ij}wherei,j≤\frac{n}{2}& Z_{ij}wherei≤\frac{n}{2}and\frac{n}{2}<j≤n\\ Z_{ij}where\frac{n}{2}<i≤nandj≤\frac{n}{2}& Z_{ij}where\frac{n}{2}<i,j≤n\end{array}\right]$

$Z=\left[\begin{array}{cc}AE+BG& AF+BH\\ CE+DG& CF+DH\\ & \end{array}\right]$

Therefore, the given property is proved.