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Help with matrix form of the imaginary unit, i 
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#1
Jun1214, 02:41 PM

P: 25

While investigating more about complex numbers today I ran across the 2x2 matrix representation of a complex number, and I was really fascinated. You can read what I read here.
As I understand it, you write z in its binomial form but instead of "1" you use the identity matrix, I, and for i you use a matrix m such that [itex]M^2=I[/itex], like with the more usual definition of i. [itex]z =x\cdot 1+y\cdot 1 \cdot i\sim z=x\cdot \begin{bmatrix}1 &0 \\ 0 & 1\end{bmatrix} +y\cdot \begin{bmatrix}1 &0 \\ 0 & 1\end{bmatrix} \begin{bmatrix}0 &1 \\ 1 & 0\end{bmatrix}=\begin{bmatrix}x &0 \\ 0 & x\end{bmatrix} +\begin{bmatrix}0 &y \\ y & 0\end{bmatrix}=\begin{bmatrix}x &y \\ y & x\end{bmatrix}[/itex] In the "normal" way (the one you learn first, at least in my case) i is defined as follows: [itex]i^{2}=1[/itex], the positive solution to the equation [itex]x^2+1=0[/itex]. This equation has two solutions, and by convention i is the positive one. If we try to solve [itex]M^2=I[/itex], then we get infinite possibilites: [itex]M^2=I \Rightarrow \begin{bmatrix}1 &0 \\ 0 &1 \end{bmatrix}=\begin{bmatrix}a &b \\ c&d \end{bmatrix}\begin{bmatrix} a&b \\ c&d \end{bmatrix}=\begin{bmatrix}a^2+bc &b(a+d) \\ c(a+d)& d^2+bc \end{bmatrix}\Rightarrow \left\{\begin{matrix}a^2+bc=1 \\ d^2+bc=1 \\ b(a+d)=0 \\ c(a+d)=0 \end{matrix}\right.[/itex] There aren't just 2 solutions now. Why is the matrix [itex]\begin{bmatrix} 0& 1\\ 1&0 \end{bmatrix}[/itex] chosen over any of the rest of the matrices that satisfy m^2=I? What is so convenient about that form over any other? Perhaps i isn't defined only as the solution to m^2=I, but if so, what am I missing? Thank you for reading. 


#2
Jun1214, 02:58 PM

P: 326

Any matrix of the form ##U^{1}\left[\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right]U##, where ##U## is a unitary matrix, can be used to represent the imaginary unit.
Mathematical groups, of which the complex numbers are one example, generally have many different representations as sets of matrices. 


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