The determinant of a rotation matrix is +1.
A rotation matrix is a special type of matrix used in linear algebra to describe rotations in Euclidean space. A key property of these matrices is related to their determinant.
Based on the provided information:
- Rotation matrices have a determinant of +1.
This property is fundamental and holds true for rotation matrices of any dimension (2D, 3D, etc.). Geometrically, the determinant represents how a linear transformation scales area (in 2D) or volume (in 3D). A determinant of +1 means the transformation preserves both the magnitude of the area/volume and the orientation of the space. Rotation does exactly this: it moves points around a center but doesn't stretch, compress, or flip the space.
Why is the Determinant +1 for Rotations?
- Orientation Preservation: Rotation does not change the "handedness" of the coordinate system. A positive determinant indicates that orientation is preserved.
- Area/Volume Preservation: Pure rotations do not stretch or shrink space. The scaling factor on area or volume is 1.
Let's compare this to other transformations:
| Transformation Type | Determinant Value | Geometric Effect | Orientation |
|---|---|---|---|
| Rotation | +1 | Preserves area/volume; no reflection | Preserved |
| Reflection | −1 | Preserves area/volume; reflection | Reversed |
| Scaling (factor k) | kn | Scales area/volume by kn | Preserved (if k>0) |
Note: 'n' is the dimension of the space.
Example: 2D Rotation Matrix
A standard 2D rotation matrix for rotating by an angle $\theta$ counterclockwise is:
$R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix}$
Let's calculate its determinant:
Determinant = $(\cos \theta \times \cos \theta) - (-\sin \theta \times \sin \theta)$
Determinant = $\cos^2 \theta - (-\sin^2 \theta)$
Determinant = $\cos^2 \theta + \sin^2 \theta$
Using the fundamental trigonometric identity, $\cos^2 \theta + \sin^2 \theta = 1$.
So, the determinant is indeed +1.
Rotation Matrices and Orthogonal Groups
Rotation matrices are part of a larger family of matrices called orthogonal matrices. Orthogonal matrices $Q$ satisfy the property $Q^T Q = I$, where $Q^T$ is the transpose and $I$ is the identity matrix. For an orthogonal matrix $Q$, the determinant can only be +1 or -1.
- Orthogonal matrices with a determinant of +1 are rotation matrices. They form a group known as the special orthogonal group, denoted SO(n) for dimension n.
- Orthogonal matrices with a determinant of −1 are reflection matrices.
As noted in the reference, "The set of all orthogonal two-dimensional matrices together with matrix multiplication form the orthogonal group: O(2)." Within O(2), rotation matrices are specifically those with determinant +1.
