When you multiply two complex numbers, you get
\begin{equation}
(a+bi)\times(c+di)=(ac-bd)+(ad+bc)i
\end{equation}
As betterexplained.com points out, this multiplication of two complex numbers can be understood as a rotation. They give an example of a ship sailing 3 units east for every 4 units north, and ask what is the effect of rotating 45 degrees counterclockwise. Using complex numbers, you can find that
\begin{equation}
(3+4i)\times(1+i)=-1+7i
\end{equation}
Therefore using complex numbers, we find that the answer we were looking for: That is, after a 45 degree rotation, the ship is sailing 1 unit west for every 7 units north.
Now as a linear algebra lover, and it seems that the use of $i$ to solve this problem is unnecessary. Rather, it seems like we can just represent what's happening here in terms of vectors in the plane, namely the vectors $(3,4)$ and $(1,1)$. Some operation will take these vectors to the vector $(-1,7)$. In general, the operation would look like this:
\begin{equation}
(a,b)*(c,d)=(ac-bd,ad+bc)
\end{equation}
My question is, can this "rotation" operation be represented some other way, i.e. in terms of matrices, and does this operation have a name? Thanks!
Best Answer
$$ \left( \begin{array}{rr} a & b \\ -b & a \end{array} \right) \left( \begin{array}{cc} c & d \\ -d & c \end{array} \right) = \left( \begin{array}{cc} ac-bd & ad+bc \\ -(ad+bc) & ac-bd \end{array} \right) $$