Chapter 40. Rotation Operators

A rotation operator is a linear operator that turns vectors around the origin while preserving lengths, angles, and orientation. In Euclidean space, rotations are represented by orthogonal matrices with determinant $1$ . Orthogonal matrices preserve inner products, and rotations form the orientation-preserving part of the orthogonal transformations.

In the plane, rotation by angle $\theta$ counterclockwise is represented by

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

For a vector

v= \begin{bmatrix} x\\ y \end{bmatrix},

the rotated vector is

R_\theta v= \begin{bmatrix} x\cos\theta-y\sin\theta\\ x\sin\theta+y\cos\theta \end{bmatrix}.

This is the standard rotation matrix for counterclockwise rotation of column vectors in the plane.

40.1 Rotation in the Plane

Let

v= \begin{bmatrix} x\\ y \end{bmatrix} \in\mathbb{R}^2.

A rotation through angle $\theta$ about the origin sends $v$ to

\begin{bmatrix} x\cos\theta-y\sin\theta\\ x\sin\theta+y\cos\theta \end{bmatrix}.

Thus the rotation operator is

R_\theta:\mathbb{R}^2\to\mathbb{R}^2,

where

R_\theta \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} x\cos\theta-y\sin\theta\\ x\sin\theta+y\cos\theta \end{bmatrix}.

This map is linear because both output coordinates are linear expressions in $x$ and $y$ .

The matrix of $R_\theta$ in the standard basis is

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

The first column is the image of $e_1$ . The second column is the image of $e_2$ :

R_\theta e_1= \begin{bmatrix} \cos\theta\\ \sin\theta \end{bmatrix}, \qquad R_\theta e_2= \begin{bmatrix} -\sin\theta\\ \cos\theta \end{bmatrix}.

The standard basis is rotated by the same angle $\theta$ .

40.2 Special Angles

Several rotations occur often.

For $\theta=0$ ,

R_0= \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} =I.

For $\theta=\frac{\pi}{2}$ ,

R_{\pi/2}= \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}.

This sends

\begin{bmatrix} x\\ y \end{bmatrix}

\begin{bmatrix} -y\\ x \end{bmatrix}.

For $\theta=\pi$ ,

R_\pi= \begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix} =-I.

This sends every vector to its negative.

For $\theta=\frac{3\pi}{2}$ ,

R_{3\pi/2}= \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix}.

This is clockwise rotation by $90^\circ$ .

40.3 Rotation Preserves Length

A rotation does not change the Euclidean length of a vector.

Let

v= \begin{bmatrix} x\\ y \end{bmatrix}.

Then

R_\theta v= \begin{bmatrix} x\cos\theta-y\sin\theta\\ x\sin\theta+y\cos\theta \end{bmatrix}.

Compute the squared length:

\|R_\theta v\|^2 = (x\cos\theta-y\sin\theta)^2 + (x\sin\theta+y\cos\theta)^2.

Expand:

=x^2\cos^2\theta-2xy\sin\theta\cos\theta+y^2\sin^2\theta +x^2\sin^2\theta+2xy\sin\theta\cos\theta+y^2\cos^2\theta.

The mixed terms cancel. Therefore

\|R_\theta v\|^2 = x^2(\cos^2\theta+\sin^2\theta) + y^2(\sin^2\theta+\cos^2\theta).

Since

\cos^2\theta+\sin^2\theta=1,

we get

\|R_\theta v\|^2=x^2+y^2=\|v\|^2.

Hence

\|R_\theta v\|=\|v\|.

40.4 Rotation Preserves Inner Products

Rotations also preserve inner products.

Let

u,v\in\mathbb{R}^2.

A rotation matrix satisfies

R_\theta^TR_\theta=I.

Therefore

(R_\theta u)\cdot(R_\theta v) = (R_\theta u)^T(R_\theta v) = u^TR_\theta^TR_\theta v = u^Tv = u\cdot v.

Since lengths and angles are determined by inner products, rotations preserve both lengths and angles. This is why rotations are rigid motions fixing the origin.

40.5 Orthogonality of Rotation Matrices

A real square matrix $Q$ is orthogonal if

Q^TQ=I.

The plane rotation matrix is orthogonal. Indeed,

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

Then

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

Multiplying gives

R_\theta^TR_\theta = \begin{bmatrix} \cos^2\theta+\sin^2\theta & -\cos\theta\sin\theta+\sin\theta\cos\theta\\ -\sin\theta\cos\theta+\cos\theta\sin\theta & \sin^2\theta+\cos^2\theta \end{bmatrix} = I.

Thus

R_\theta^{-1}=R_\theta^T.

The inverse of a rotation through $\theta$ is rotation through $-\theta$ :

R_\theta^{-1}=R_{-\theta}.

40.6 Determinant

The determinant of a plane rotation is

\det(R_\theta) = \det \begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix}.

Compute:

\det(R_\theta) = \cos^2\theta+\sin^2\theta = 1.

Thus a rotation preserves oriented area.

Orthogonal transformations have determinant either $1$ or $-1$ . Those with determinant $1$ preserve orientation. Those with determinant $-1$ reverse orientation. Rotations are the orientation-preserving orthogonal transformations.

40.7 Composition of Plane Rotations

Rotations in the plane compose by adding angles.

Let

R_\alpha= \begin{bmatrix} \cos\alpha & -\sin\alpha\\ \sin\alpha & \cos\alpha \end{bmatrix}, \qquad R_\beta= \begin{bmatrix} \cos\beta & -\sin\beta\\ \sin\beta & \cos\beta \end{bmatrix}.

Then

R_\alpha R_\beta = \begin{bmatrix} \cos(\alpha+\beta) & -\sin(\alpha+\beta)\\ \sin(\alpha+\beta) & \cos(\alpha+\beta) \end{bmatrix}.

Hence

R_\alpha R_\beta=R_{\alpha+\beta}.

This follows from the angle addition identities.

Because real number addition is commutative,

R_\alpha R_\beta=R_\beta R_\alpha

for plane rotations about the origin. This commutativity is special. General matrices do not commute, and rotations in three dimensions generally do not commute.

40.8 Powers of a Rotation

Since composition adds angles,

R_\theta^k=R_{k\theta}

for every integer $k\geq 0$ .

Also,

R_\theta^{-1}=R_{-\theta},

so the formula extends to negative integers:

R_\theta^k=R_{k\theta}

for all integers $k$ .

If $\theta$ is a rational multiple of $2\pi$ , then some power of $R_\theta$ is the identity. For example,

R_{\pi/2}^4=R_{2\pi}=I.

If $\theta$ is not a rational multiple of $2\pi$ , then the powers never return exactly to the identity, although the rotated directions may come arbitrarily close.

40.9 Eigenvalues in the Plane

A real plane rotation usually has no real eigenvectors.

Suppose

R_\theta v=\lambda v

for some nonzero real vector $v$ . This means the rotation leaves the line spanned by $v$ invariant. For most angles, a rotation changes every line through the origin.

The characteristic polynomial of

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

p(\lambda)=\lambda^2-2(\cos\theta)\lambda+1.

The roots are

\lambda=\cos\theta\pm i\sin\theta.

Using Euler notation, these are

\lambda=e^{i\theta}

and

\lambda=e^{-i\theta}.

Thus over $\mathbb{C}$ , every plane rotation is diagonalizable unless the roots coincide. Over $\mathbb{R}$ , it has real eigenvectors only for special angles.

When $\theta=0$ , both eigenvalues are $1$ . When $\theta=\pi$ , both eigenvalues are $-1$ . For other angles, the eigenvalues are nonreal complex conjugates.

40.10 Trace and Angle

For a plane rotation,

\operatorname{tr}(R_\theta)=2\cos\theta.

Indeed,

\operatorname{tr} \begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix} = \cos\theta+\cos\theta = 2\cos\theta.

Thus the angle is determined up to sign by the trace:

\cos\theta=\frac{\operatorname{tr}(R_\theta)}{2}.

The trace cannot distinguish $\theta$ from $-\theta$ , because

\cos(-\theta)=\cos\theta.

The sign of the sine term determines the orientation of the rotation.

40.11 Rotations as Complex Multiplication

The plane $\mathbb{R}^2$ can be identified with the complex plane $\mathbb{C}$ by

\begin{bmatrix} x\\ y \end{bmatrix} \leftrightarrow z=x+iy.

Multiplication by

e^{i\theta}=\cos\theta+i\sin\theta

gives

e^{i\theta}z = (\cos\theta+i\sin\theta)(x+iy).

Expanding,

e^{i\theta}z = (x\cos\theta-y\sin\theta) + i(x\sin\theta+y\cos\theta).

This is exactly the matrix formula for $R_\theta$ .

Thus plane rotation is complex multiplication by a unit complex number.

This viewpoint explains why rotations in the plane commute:

e^{i\alpha}e^{i\beta}=e^{i(\alpha+\beta)}=e^{i\beta}e^{i\alpha}.

40.12 The Rotation Group in the Plane

The set of all plane rotations about the origin is

SO(2)=\{R\in\mathbb{R}^{2\times 2}:R^TR=I,\ \det(R)=1\}.

This is the special orthogonal group in dimension $2$ .

It is a group under matrix multiplication.

The identity is

R_0=I.

The inverse of $R_\theta$ is

R_{-\theta}.

The product of two rotations is another rotation:

R_\alpha R_\beta=R_{\alpha+\beta}.

Every element of $SO(2)$ has the form

\begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix}

for some real $\theta$ .

40.13 Rotations in Three Dimensions

A rotation in $\mathbb{R}^3$ is a linear transformation that preserves lengths, preserves angles, preserves orientation, and fixes the origin.

Its matrix $R$ satisfies

R^TR=I

and

\det(R)=1.

Thus

R\in SO(3).

Unlike plane rotations, rotations in $\mathbb{R}^3$ usually have an axis. Euler’s rotation theorem states that every rotation of three-dimensional space has a fixed axis, and the rotation acts as an ordinary plane rotation on the perpendicular plane.

40.14 Rotation About a Coordinate Axis

Rotation about the $z$ -axis by angle $\theta$ has matrix

R_z(\theta)= \begin{bmatrix} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{bmatrix}.

It rotates the $xy$ -coordinates and leaves the $z$ -coordinate fixed.

Rotation about the $x$ -axis has matrix

R_x(\theta)= \begin{bmatrix} 1 & 0 & 0\\ 0 & \cos\theta & -\sin\theta\\ 0 & \sin\theta & \cos\theta \end{bmatrix}.

Rotation about the $y$ -axis has matrix

R_y(\theta)= \begin{bmatrix} \cos\theta & 0 & \sin\theta\\ 0 & 1 & 0\\ -\sin\theta & 0 & \cos\theta \end{bmatrix}.

Each matrix is orthogonal and has determinant $1$ .

40.15 Noncommutativity in Three Dimensions

Rotations in $\mathbb{R}^3$ generally do not commute.

Let $R_x(\alpha)$ be rotation about the $x$ -axis and $R_y(\beta)$ be rotation about the $y$ -axis. In general,

R_x(\alpha)R_y(\beta)\neq R_y(\beta)R_x(\alpha).

Geometrically, turning an object about one axis and then another usually gives a different final orientation from doing the turns in the opposite order.

This is a major difference between $SO(2)$ and $SO(3)$ . The group $SO(2)$ is abelian. The group $SO(3)$ is nonabelian.

40.16 Axis and Angle in Three Dimensions

A nonidentity rotation in $\mathbb{R}^3$ has an axis. Vectors on the axis are fixed:

Rv=v.

Thus the axis is the eigenspace for eigenvalue $1$ :

\ker(R-I).

On the plane perpendicular to the axis, the operator acts as a two-dimensional rotation.

For a three-dimensional rotation matrix $R$ , the trace determines the rotation angle $\theta$ by

\operatorname{tr}(R)=1+2\cos\theta.

Hence

\cos\theta=\frac{\operatorname{tr}(R)-1}{2}.

The axis is found by solving

(R-I)v=0.

This gives the fixed direction.

40.17 Rodrigues’ Rotation Formula

Let $u\in\mathbb{R}^3$ be a unit vector. Rotation through angle $\theta$ about the axis spanned by $u$ is given by Rodrigues’ formula:

R(v)=v\cos\theta+(u\times v)\sin\theta+u(u\cdot v)(1-\cos\theta).

This formula decomposes $v$ into a component parallel to the axis and a component perpendicular to the axis.

The parallel part is unchanged. The perpendicular part is rotated in the plane normal to $u$ .

In matrix form, let

[u]_\times= \begin{bmatrix} 0 & -u_3 & u_2\\ u_3 & 0 & -u_1\\ -u_2 & u_1 & 0 \end{bmatrix}.

Then

R=I\cos\theta+(1-\cos\theta)uu^T+\sin\theta [u]_\times.

This is the axis-angle representation of a three-dimensional rotation.

40.18 Skew-Symmetric Generators

Rotations are generated infinitesimally by skew-symmetric matrices.

A matrix $A$ is skew-symmetric if

A^T=-A.

If $A$ is skew-symmetric, then

e^{tA}

is orthogonal for every real $t$ .

To see this, define

Q(t)=e^{tA}.

Then

Q(t)^TQ(t)=e^{tA^T}e^{tA}=e^{-tA}e^{tA}=I.

Thus $Q(t)$ is orthogonal.

In the plane,

J= \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}

is skew-symmetric, and

e^{\theta J} = \begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix} = R_\theta.

This connects rotations with matrix exponentials and Lie theory.

40.19 Rotations and Reflections

Rotations and reflections are both orthogonal transformations, but they differ by determinant.

A reflection has determinant $-1$ in the hyperplane case. A rotation has determinant $1$ .

In $\mathbb{R}^2$ , every orthogonal matrix with determinant $1$ is a rotation. Every orthogonal matrix with determinant $-1$ is a reflection across a line through the origin.

Thus

O(2)

splits into two parts:

SO(2)

for rotations, and the determinant $-1$ part for reflections.

In higher dimensions, determinant $1$ distinguishes orientation-preserving orthogonal transformations, though their geometry may involve rotations in several orthogonal planes.

40.20 Higher-Dimensional Rotations

In $\mathbb{R}^n$ , a rotation is usually defined as an element of

SO(n)=\{R\in\mathbb{R}^{n\times n}:R^TR=I,\ \det(R)=1\}.

Such a matrix preserves inner products and orientation.

In dimensions greater than $3$ , a rotation need not have a single axis in the familiar three-dimensional sense. Instead, an orthogonal transformation can be decomposed, after choosing a suitable orthonormal basis, into blocks:

\begin{bmatrix} \cos\theta_j & -\sin\theta_j\\ \sin\theta_j & \cos\theta_j \end{bmatrix}

on mutually orthogonal two-dimensional planes, together with possible $1\times 1$ blocks equal to $1$ or $-1$ .

For proper rotations, the determinant of all blocks together is $1$ .

This block structure generalizes the plane rotation matrix.

40.21 Rotations in Computation

Rotation operators appear throughout computation.

In computer graphics, rotations orient cameras, objects, and coordinate frames. In robotics, they describe rigid-body orientation. In numerical linear algebra, orthogonal transformations help reduce matrices while preserving norms. In differential equations, rotations describe periodic and oscillatory behavior.

Because rotation matrices are orthogonal, they are numerically well-conditioned:

\|Rx\|=\|x\|.

Thus applying a rotation does not magnify Euclidean error. This is one reason orthogonal transformations are preferred in stable numerical algorithms.

40.22 Summary

A rotation operator is a linear transformation that preserves lengths, angles, and orientation.

In the plane, rotation by angle $\theta$ has matrix

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

It satisfies

R_\theta^TR_\theta=I, \qquad \det(R_\theta)=1, \qquad R_\theta^{-1}=R_{-\theta}.

Plane rotations compose by adding angles:

R_\alpha R_\beta=R_{\alpha+\beta}.

In three dimensions, a rotation matrix lies in

SO(3),

has determinant $1$ , and, by Euler’s rotation theorem, has an axis of rotation. The axis is the eigenspace for eigenvalue $1$ .

In higher dimensions, rotations are elements of

SO(n).

They preserve the inner product and orientation, and they can be studied through orthogonal block decompositions.

Rotation operators are central examples of linear maps because they combine geometry, algebra, groups, eigenvalues, matrix exponentials, and numerical stability.