Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ 
Possible Duplicate:
Relation of this antisymmetric matrix $r = \!\left(\begin{smallmatrix}0 &1\\-1 & 0\end{smallmatrix}\right)$ to $i$

On Wikipedia, it says that:

Matrix representation of complex numbers
Complex numbers $z=a+ib$ can also be represented by $2\times2$ matrices that have the following form: $$\pmatrix{a&-b\\b&a}$$

I don't understand why they can be represented by these matrices or where these matrices come from.
 A: The matrix rep of $\rm\:\alpha = a+b\,{\it i}\:$ is simply the matrix representation of the $\:\Bbb R$-linear map $\rm\:x\to \alpha\, x\:$ viewing $\,\Bbb C\cong \Bbb R^2$ as vector space over $\,\Bbb R.\,$ Computing the coefficients of $\,\alpha\,$ wrt to the basis $\,[1,\,{\it i}\,]^T\:$
$$\rm (a+b\,{\it i}\,) \left[ \begin{array}{c} 1 \\ {\it i} \end{array} \right] 
\,=\, \left[\begin{array}{r}\rm a+b\,{\it i}\\\rm -b+a\,{\it i} \end{array} \right] 
\,=\, \left[\begin{array}{rr}\rm a &\rm b\\\rm -b &\rm a \end{array} \right] 
\left[\begin{array}{c} 1 \\ {\it  i} \end{array} \right]$$
As above, any ring may be viewed as a ring of linear maps on its additive group (the so-called left-regular representation). Informally, simply view each element of the ring as a $1\!\times\! 1$ matrix, with the usual matrix operations. This is a ring-theoretic analog of the Cayley representation of a group via permutations on its underlying set, by viewing each $\,\alpha\,$ as a permutation $\rm\,x\to\alpha\,x.$
When, as above, the ring has the further structure of an $\rm\,n$-dimensional vector space over a field, then, wrt a basis of the vector space, the linear maps $\rm\:x\to \alpha\, x\:$ are representable as $\rm\,n\!\times\!n\,$ matrices; e.g.  any algebraic field extension of degree $\rm\,n.\,$ Above is the special case $\rm n=2.$
A: No one seems to have mentioned it explicitly, so I will. The matrix $J = \left( \begin{smallmatrix} 0 & -1\\1 & 0 \end{smallmatrix} \right)$ satisfies $J^{2} = -I,$ where $I$ is the $2 \times 2$ identity matrix (in fact, this is because $J$ has eigenvalues $i$ and $-i$, but let us put that aside for one moment). Hence there really is no difference between the matrix $aI + bJ$ and the complex number $a +bi.$
A: The matrices $I=\begin{bmatrix}1&0\\0&1\end{bmatrix}$ and $J=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ commute (everything commutes with $I$), and $J^2=-I$. Everything else follows from the standard properties (associativity, commutativity, distributivity, etc.) that matrix operations have.
Thus, $aI+bJ=\begin{bmatrix}a&-b\\b&a\end{bmatrix}$ behaves exactly like $a+bi$ under addition, multiplication, etc.
A: Look at the arithmetic operations and their actions.  With + and *, these matrices form a field.  And we have the isomorphism
$$a + ib \mapsto \left[\matrix{a&-b\cr b &a}\right].$$
A: What I think of a complex number is a scaling and 2D rotation operation, where the absolute value $r$ is scaling factor, and the phase $\theta$ is the rotation angle.
The same operation can be described by scalar multiplication of a rotation matrix as $$r\begin{pmatrix}\cos \theta & -\sin \theta \\  \sin \theta & \cos \theta \end{pmatrix}$$
Since $r e^{i\theta}=r\cos \theta + ir \sin \theta = a +ib$, we have $$a +ib = \begin{pmatrix}a & -b \\  b & a \end{pmatrix}$$
A: Since you put the tag quaternions, let me say a bit more about performing identifications like that: 
Recall the quaternions $\mathcal{Q}$ is the group consisting of elements $\{\pm1, \pm \hat{i}, \pm \hat{j}, \pm \hat{k}\}$ equipped with multiplication that satisfies the rules according to the diagram
$$\hat{i} \rightarrow \hat{j} \rightarrow \hat{k}.$$
Now what is more interesting is that you can let $\mathcal{Q}$ become a four dimensional real vector space with basis $\{1,\hat{i},\hat{j},\hat{k}\}$ equipped with an $\Bbb{R}$ - bilinear multiplication map that satisfies the rules above. You can also define the norm of a quaternion $a + b\hat{i} + c\hat{j} + d\hat{k}$ as 
$$||a + b\hat{i} + c\hat{j} + d\hat{k}|| = a^2 + b^2 + c^2 + d^2.$$
Now if you consider $\mathcal{Q}^{\times}$, the set of all unit quaternions you can identify $\mathcal{Q}^{\times}$ with $\textrm{SU}(2)$ as a group and as a topological space. How do we do this identification? Well it's not very hard. Recall that 
$$\textrm{SU}(2) = \left\{ \left(\begin{array}{cc} a + bi & -c + di \\ c + di & a-bi \end{array}\right) |\hspace{3mm} a,b,c,d \in \Bbb{R}, \hspace{3mm} a^2 + b^2 + c^2 + d^2 = 1    \right\}.$$
So you now make an ansatz (german for educated guess) that the identification we are going to make is via the map $f$ that sends a quaternion $a + b\hat{i} + c\hat{j} + d\hat{k}$ to the matrix $$\left(\begin{array}{cc} a + bi & -c + di \\ c + di & a-bi \end{array}\right).$$
It is easy to see that $f$  is a well-defined group isomorphism by an algebra bash and it is also clear that $f$ is a homeomorphism. In summary, the point I wish to make is that these identifications give us a useful way to interpret things. For example, instead of interpreting $\textrm{SU}(2)$ as boring old matrices that you say "meh" to you now have a geometric understanding of what $\textrm{SU}(2)$. You can think about each matrix as being a point on the sphere $S^3$ in 4-space! How rad is that?
On the other hand when you say $\Bbb{R}^4$ has now basis elements consisting of $\{1,\hat{i},\hat{j},\hat{k}\}$, you have given $\Bbb{R}^4$ a multiplication structure and it becomes not just an $\Bbb{R}$ - module but a module over itself. 
A: As to the "where did it come from?", rather than verifying that it does work: this is a special case of a "rational representation" of a bigger collection of "numbers" as matrices with entries in a smaller collection. ("Fields" or "rings", properly, but it's not clear what our context here is.)
That is, the collection of complex numbers is a two-dimensional real vector space, and multiplication by $a+bi$ is a real-linear map of $\mathbb C$ to itself, so, with respect to any $\mathbb R$-basis of $\mathbb C$, there'll be a corresponding matrix. For example, with $\mathbb R$-basis $e_1=1,\,e_2=i$, 
$$
(a+bi)\cdot e_1 = a+bi = ae_1+be_2
\hskip40pt
(a+bi)\cdot e_2 = (a+bi)i = -b+ai = -be_1+ae_2
$$
So
$$
\pmatrix{e_1 \cr e_2}\cdot (a+bi) \;=\; \pmatrix{a & b \cr -b & a}\pmatrix{e_1\cr e_2}
$$
Oop, I guessed wrong, and got the $b$ and $-b$ interchanged. Maybe using $e_2=-i$ instead will work... :)
But this is the way one finds such representations.
A: I had something written up on this lying around. The $-$ sign is off, but it's more or less the same, I hope it helps.
Let $M$ denote the set of such matrices. Define a function $\phi\colon M\to\mathbb{C}$ by
$$
\begin{pmatrix} \alpha & \beta \\ -\beta & \alpha\end{pmatrix}\mapsto \alpha+i\beta.
$$
Note that this function has inverse $\phi^{-1}$ defined by $\alpha+i\beta\mapsto\begin{pmatrix} \alpha & \beta \\ -\beta & \alpha\end{pmatrix}$. This function is well defined, since $\alpha+i\beta=\gamma+i\delta$ if and only if $\alpha=\gamma$ and $\beta=\delta$, and thus it is never the case that a complex number can be written in two distinct ways with different real part and different imaginary part. So $\phi$ is invertible.
Now let 
$$
A=\begin{pmatrix} \alpha & \beta \\ -\beta & \alpha\end{pmatrix},\qquad 
B=\begin{pmatrix} \gamma & \delta \\ -\delta & \gamma\end{pmatrix}.
$$
Then
$$
\phi(A+B)=\phi\begin{pmatrix} \alpha+\gamma & \beta+\delta \\ -\beta-\delta & \alpha+\delta\end{pmatrix}=(\alpha+\gamma)+i(\beta+\delta)=(\alpha+i\beta)+(\gamma+i\delta)=\phi(A)+\phi(B).
$$
Also,
$$
\phi(AB)=\phi\begin{pmatrix} \alpha\gamma-\beta\delta & \alpha\delta+\beta\gamma \\ -\beta\gamma-\alpha-\delta & -\beta\delta+\alpha\gamma\end{pmatrix}=(\alpha\gamma-\beta\delta)+i(\alpha\delta+\beta\gamma)=(\alpha+i\beta)(\gamma+i\delta)=\phi(A)\phi(B).
$$
So $\phi$ respects addition and multiplication. Lastly, $\phi(I_2)=1$, so $\phi$ also respects the multiplicative identity. Hence $\phi$ is a field isomorphism, so $M$ and $\mathbb{C}$ are isomorphic as fields.
