Matrix Lie Groups Explained: Rotations in Plane and Complex Numbers as Matrices

Mathematics often hides deep connections in plain sight. A rotation in a plane, a complex number, and a special matrix—at first glance, they seem like unrelated objects. Yet, they are facets of the same elegant structure: matrix Lie groups.

In this post, we explore how rotations in the plane form the Lie group $SO(2)$ , and how complex numbers naturally arise as $2\times 2$ real matrices, revealing a beautiful bridge between algebra and geometry.

Introduction

A Lie group is, simply put, a group that is also a smooth manifold where the group operations are compatible with the smooth structure. This remarkable marriage allows us to perform calculus on groups. We are familiar with the idea that (Discrete) groups capture discrete symmetries. Lie groups, on the other hand, capture continuous symmetries. This can be a topic for a later blog post.

For the time being, we explore the most tangible examples of Lie groups which are matrix Lie groups — groups of matrices under multiplication that also form smooth manifolds.

In this post, we will:

Understand rotations in the $2$ D plane.
Explore the rotation group $SO(2)$ .
Discover how complex numbers can be realized as $2\times 2$ real matrices.

Rotations in the Plane

Imagine rotating the plane $\mathbb R^2$ counterclockwise by an angle $\theta$ about the origin.

The standard basis vectors transform as follows:
$\text{\space}e_1=\begin{bmatrix}1\\0\end{bmatrix}\mapsto\begin{bmatrix}\cos\theta\\\sin\theta\end{bmatrix}\\\\\text{and\space}e_2=\begin{bmatrix}0\\1\end{bmatrix}\mapsto\begin{bmatrix}-\sin\theta\\\cos\theta\end{bmatrix}$

Therefore, the rotation is the linear operator $T_\theta$ given by the matrix:

$R_\theta=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.$

with respect to the standard ordered basis $\{e_1,e_2\}$ .

These matrices satisfy the familiar properties:

$R_\theta\cdot R_\phi=R_{\theta+\phi}\text{\space and\space}R_\theta\cdot R_{-\theta}=R_{-\theta}\cdot R_\theta=R_0=I$

The collection of all such rotations $SO(2)=\{R_\theta:0\le\theta<2\pi\}$ forms a group under matrix multiplication, called the Special Orthogonal Group of order $2$ .

The Complex View of Rotation

Rotations in the plane have a dual identity—they are also multiplications by complex numbers on the unit circle.

Identify a complex number $x+iy\in\mathbb{C}$ with the column vector $\begin{bmatrix}x\\y\end{bmatrix}\in \mathbb R^2$ .
Multiplying $z=x+iy$ by $e^{i\theta}=\cos\theta+i\sin\theta$ gives:

$z\cdot e^{i\theta}=(x+iy)(\cos\theta+i\sin\theta)\\=(x\cos\theta-y\sin\theta)+i(x\sin\theta+y\cos\theta)\\=\begin{bmatrix}x\cos\theta-y\sin\theta\\x\sin\theta+y\cos\theta\end{bmatrix}\text{\space Under the canonical identification between\space}\mathbb C\text{\space and\space}\mathbb R^2\\=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}\cdot\begin{bmatrix}x\\y\end{bmatrix}\\=R_\theta\cdot\begin{bmatrix}x\\y\end{bmatrix}$

Two worlds, one rotation: Whether you think in matrices or complex numbers, it’s the same transformation.

The Rotation Group $SO(2)$

The group $SO(2)$ consists of all rotations in $\mathbb R^2$ : $SO(2)=\{R\in GL(2,\mathbb{R})\mid R\cdot R^T=I=R^T\cdot R,\,\det(R)=1\}$ .

$R\cdot R^T=R^T\cdot R=I$ ensures orthogonality.
$\det(R)=+1$ ensures no reflections, only rotations.

This is our first example of a matrix Lie group.

Topological Properties of $SO(2)$

Let’s explore the topological properties of $SO(2)$ . Note that $SO(2)\subset M(2,\mathbb R)$ , and the latter can be naturally identified with $\mathbb R^4$ , which is a normed space with Euclidean norm. Thus, it makes sense talk about the Euclidean normed topology on $M(2,\mathbb R)$ , and consequently the subspace topology on $SO(2)$ induced from this Euclidean topology. We would first explore the compactness properties of $SO(2)$ .

Take a general element $R_\theta\in SO(2)$ , the Euclidean norm of $R_\theta$ would be :
$\|R_\theta\|_{E}=\sqrt{\cos^2\theta+(-\sin\theta)^2+\sin^2\theta+\cos^2\theta}\\\text{\space\space\space\space\space\space\space\space\space\space\space\space\space}=\sqrt{2\cdot(\cos^2\theta+\sin^2\theta)}=\sqrt{2}$
and thus, $\|R_\theta\|_E\leq\sqrt 2,\text{\space for every\space}\theta$ . Thus, $SO(2)$ is a bounded subset of $M(2,\mathbb R)$ .

We will see that $SO(2)$ is closed in $M(2,\mathbb R)$ . To see this, define, $f:M(2,\mathbb R)\to M(2,\mathbb R)$ by $f(A)=A^T\cdot A$ and $g:M(2,\mathbb R)\to\mathbb R$ by $g(A)=\det(A)$ . Clearly, we can write: $SO(2)=\{A\in M(2,\mathbb R):A^T\cdot A=I\}\cap\{A\in M(2,\mathbb R):\det(A)=1\}\\\\=\{A\in M(2,\mathbb R):f(A)=I\}\cap\{A\in M(2,\mathbb R):g(A)=1\}=f^{-1}(I)\cap g^{-1}(1)$ . Also, note that both $f\text{\space and\space}g$ are continuous. We know that the inverse image of closed sets is closed under continuous maps, and $f^{-1}(I)\text{\space and\space}g^{-1}(1)$ are inverse images of the closed sets $\{I\}\text{\space and\space}\{1\}$ under the continuous maps $f,g$ respectively. Thus, $SO(2)=f^{-1}(I)\cap g^{-1}(1)$ , intersection of two closed subsets of $M(2,\mathbb R)$ is closed in $M(2,\mathbb R)$ .

$SO(2)\subset M(2,\mathbb R)$ is both closed and bounded, and hence it is compact, by Heine-Borel Theorem.

We will now see that $SO(2)$ is a path-connected topological space. Take two arbitrary elements $R_\theta,R_\phi\in SO(2)$ with $\theta,\phi\in[0,2\pi)$ . Consider the path $\Theta(t)=(1-t)\theta+t\phi,\text{\space with\space}t\in[0,1]$ . $\Theta$ is a path in $[0,2\pi)$ and then consider $\gamma(t):=R_{\Theta(t)}\in SO(2)$ , where $\gamma:[0,1]\to SO(2)$ is a path satisfying $\gamma(0)=R_\theta,\gamma(1)=R_\phi$ . Thus, $\gamma$ is a path in $SO(2)$ joining $R_\theta\text{\space and\space}R_\phi$ , proving that $SO(2)$ is path-connected.
Thus, we saw that $SO(2)$ is both compact and path-connected.

Complex Numbers as Real $2\times 2$ Matrices

A complex number $a+ib$ can be represented as the real $2\times 2$ matrix: $\begin{bmatrix}a&-b\\b&a\end{bmatrix}$ .

Why is this a natural identification? Because matrix multiplication of this form mirrors complex multiplication:

$(a+ib)(c+id)=(ac-bd)+i(ad+bc)\\\\\begin{bmatrix}a&-b\\b&a\end{bmatrix}\begin{bmatrix}c&-d\\d&c\end{bmatrix}=\begin{bmatrix}ac-bd&-(ad+bc)\\ad+bc&ac-bd\end{bmatrix}.$ .

They are the same operation, under the identification: $a+ib\quad\longleftrightarrow\quad\begin{bmatrix}a&-b\\b&a\end{bmatrix}.$

Notably, the imaginary unit corresponds to: $i\quad\longleftrightarrow\quad J=\begin{bmatrix}0&-1\\1&0\end{bmatrix},\qquad J^2=-I.$

Modulus and Determinant

This representation beautifully ties algebra to geometry:

$\det\begin{bmatrix}a&-b\\b&a\end{bmatrix}=a^2+b^2=|a+ib|^2.$
$\det(AB)=\det(A)\det(B)$ mirrors $|z_1 z_2|^2=|z_1|^2|z_2|^2$
$\begin{bmatrix}a&-b\\b&a\end{bmatrix}^{-1}=\frac{1}{a^2+b^2}\begin{bmatrix}a&b\\-b&a\end{bmatrix}$ mirrors $(a+ib)^{-1}=\frac{1}{a+ib}=\frac{a-ib}{a^2+b^2}$

The determinant acts as the square of the complex modulus.

The Two Squares Formula

I am tempted to mention a classical application of norm-squared at this point. From the complex multiplication rule: $(a+ib)\cdot(c+id)=(ac-bd)+i(ad+bc)$ , by applying modulus and squaring both sides, we get $(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2,$ and this give us a classic number theory result :
Given $a,b,c,d\in\mathbb R$ there exist $\alpha,\beta\in\mathbb R$ such that $(a^2+b^2)\cdot(c^2+d^2)=\alpha^2+\beta^2$ .

showing that the product of two sums of two squares is again a sum of two squares.

The Three Squares Problem

Curiously, the magic fails for three squares: $(a^2+b^2+c^2)(x^2+y^2+z^2),$

cannot always be written as a sum of three squares. This is a deep number-theoretic fact that contrasts sharply with the two-squares case.

Conclusion

Matrices, complex numbers, and Lie groups are all part of the same story. Linear algebra, geometry, and number theory meet in the humble $2\times 2$ matrix.

In the next post, we will push this journey further, exploring quaternions and their complex matrix representation, four square formula etc. Stay tuned!

Manifolds Unfolded