Complex numbers as matrices

ℂ (the complex numbers) is isomorphic to the set of $2 \times 2$ real matrices of the form

$$ M(a,b) =  \begin{bmatrix} a & b \\ -b & a \end{bmatrix}. $$

Tell me more about this

Nice — this is a very natural and useful way to see $\mathbb C$ as a real algebra of $2\times2$ matrices. I'll walk through the key facts, show why the map is an isomorphism, give formulas (inverse, determinant, eigenvalues, exponential), and point out the geometric picture.

---

The map and why it is an isomorphism

Define

$$ \varphi:\mathbb C\to M_2(\mathbb R),\qquad \varphi(a+bi)=M(a,b)=\begin{bmatrix} a & b\\ -b & a\end{bmatrix}. $$

1. Additive and multiplicative homomorphism. For $z=a+bi,\; w=c+di$,

   $$ \varphi(z+w)=M(a+c,\;b+d)=M(a,b)+M(c,d), $$

   and a direct matrix multiplication gives

   $$ M(a,b)M(c,d)=M(ac-bd,\;ad+bc)=\varphi((a+bi)(c+di)). $$

   So $\varphi$ preserves addition and multiplication.

2. Bijective. The map is clearly linear over $\mathbb R$ and $\varphi(a+bi)=0$ iff $a=b=0$, so it is injective; any matrix of the form above comes from exactly one complex number, so $\varphi$ is bijective onto the subset $\{M(a,b):a,b\in\mathbb R\}$. Hence $\varphi$ is an isomorphism of real algebras.

Thus this subset of $2\times2$ real matrices is a copy (a real-algebra embedding) of $\mathbb C$.

---

Algebraic identities (useful formulas)

Write $I=\begin{bmatrix}1&0\\0&1\end{bmatrix}$ and $J=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$. Then

$$ M(a,b)=aI+bJ,\qquad J^2=-I. $$

Because $J^2=-I$, the algebra generated by $I$ and $J$ is isomorphic to $\mathbb C\simeq\{a+ b i\}$ via $i\mapsto J$.

Determinant and trace.

$$ \det M(a,b)=a^2+b^2=|a+bi|^2,\qquad \operatorname{tr} M(a,b)=2a=2\operatorname{Re}(a+bi). $$

Inverse. If $a+bi\neq0$,

$$ M(a,b)^{-1}=\frac{1}{a^2+b^2}\,M(a,-b). $$

This matches the complex formula $(a+bi)^{-1}=\dfrac{a-bi}{a^2+b^2}$.

Transpose and conjugation.

$$ M(a,b)^T=M(a,-b), $$

so matrix transpose corresponds to complex conjugation $a+bi\mapsto a-bi$.

Orthogonality / rotations. If $a^2+b^2=1$ then $M(a,b)$ is orthogonal (indeed $M M^T=I$) and $\det=1$; these matrices represent rotations in the plane (they form a copy of the unit circle $S^1\subset\mathbb C$).

---

Eigenvalues and diagonalization

The characteristic polynomial is

$$ \lambda^2-2a\lambda+(a^2+b^2)=0, $$

so the (complex) eigenvalues are

$$ \lambda=a\pm ib, $$

i.e. the eigenvalues are exactly the complex number and its conjugate. Over $\mathbb R$ the matrix is not diagonalizable unless $b=0$, but over $\mathbb C$ it diagonalizes in the usual way.

---

Exponentials and analytic functions

Because $\varphi$ is an $\mathbb R$-algebra isomorphism, any real-coefficient power series evaluated at a complex number corresponds to evaluating the same power series at the matrix. In particular, for $z=a+ib$,

$$ \exp\big(M(a,b)\big)=M\big(\operatorname{Re}(e^z),\;\operatorname{Im}(e^z)\big) = M(e^a\cos b,\; e^a\sin b). $$

So matrix exponentiation corresponds exactly to the complex exponential.

---

Geometric interpretation

Think of $\mathbb R^2$ as the real plane. Multiplication by a complex number $z=re^{i\theta}$ is the linear map “scale by $r$” and “rotate by $\theta$”. Under $\varphi$, the matrix $M(a,b)$ implements that linear map on $\mathbb R^2$. Thus this embedding identifies $\mathbb C$ with the algebra of linear maps of $\mathbb R^2$ that are $\mathbb R$-linear and commute with the complex-structure operator $J$ (i.e. are of the form $aI+bJ$).

---

A concrete example

$(1+2i)(3+4i)=-5+10i$. Matrix side:

$$ M(1,2)M(3,4)=M(-5,10)=\begin{bmatrix}-5 & 10\\ -10 & -5\end{bmatrix}. $$

Also $\det M(1,2)=1^2+2^2=5$, and

$$ M(1,2)^{-1}=\frac{1}{5}M(1,-2)=\frac{1}{5}\begin{bmatrix}1 & -2\\ 2 & 1\end{bmatrix}. $$

---

Generalizations & remarks

• This exhibits $\mathbb C$ as a 2-dimensional real algebra inside $M_2(\mathbb R)$.
• The same idea generalizes: quaternions $\mathbb H$ can be represented inside $M_4(\mathbb R)$.
• Any real $2$-dimensional algebra containing an element whose square is $-1$ is (up to isomorphism) the complex numbers; here $J$ plays that role.

---

If you like, I can: (1) show a short proof that every matrix commuting with $J$ has the form $aI+bJ$; (2) draw the action of $M(a,b)$ on some sample vectors to illustrate the rotation/scale; or (3) show how this matrix representation relates to the usual complex-to-real 2×2 matrix $\begin{bmatrix}a&-b\\ b&a\end{bmatrix}$ (they are the same up to a simple change of coordinates). Which would you prefer?