A rotation matrix in 3D space is a 3x3 matrix with 9 entries that must satisfy specific mathematical conditions to represent a pure rotation. These conditions constrain the possible values of the matrix entries.
Understanding Rotation Matrix Constraints
The constraints placed upon a rotation matrix ensure that it preserves vector lengths and the angles between vectors, which are fundamental properties of rotations. These constraints mean that the matrix represents a rigid transformation that only changes the orientation of a vector or coordinate system, not its size or shape.
Specifically, for a 3x3 matrix to be a rotation matrix, its columns (and rows) must form an orthonormal basis. This property translates into a set of 6 constraints on the 9 entries of the matrix.
The Six Key Constraints
According to the provided reference, the 9 entries of the matrix must be subject to 6 constraints. These constraints relate to the properties of the matrix's column vectors:
Unit Vector Constraints (3 Constraints)
Each column vector of the rotation matrix must be a unit vector. This means the length (or Euclidean norm) of each column vector must be equal to 1. If a column vector is represented as $\mathbf{c} = [c_x, c_y, c_z]^T$, its norm is $|\mathbf{c}| = \sqrt{c_x^2 + c_y^2 + c_z^2}$.
- Constraint 1: The first column vector is a unit vector (its norm is 1).
- Constraint 2: The second column vector is a unit vector (its norm is 1).
- Constraint 3: The third column vector is a unit vector (its norm is 1).
Orthogonality Constraints (3 Constraints)
Any two distinct column vectors of the rotation matrix must be orthogonal to each other. This means their dot product must be zero. If two column vectors are $\mathbf{c}_i$ and $\mathbf{c}_j$, their dot product is $\mathbf{c}_i \cdot \mathbf{c}j = c{ix}c{jx} + c{iy}c{jy} + c{iz}c_{jz}$.
- Constraint 4: The dot product of the first and second column vectors is zero.
- Constraint 5: The dot product of the first and third column vectors is zero.
- Constraint 6: The dot product of the second and third column vectors is zero.
In summary, these 6 constraints mean that the three column vectors are orthogonal to each other and each has a length of one.
Practical Implications
These six constraints combined are equivalent to stating that the transpose of the matrix is equal to its inverse ($R^T = R^{-1}$), meaning the matrix is orthogonal ($R^T R = I$, where $I$ is the identity matrix). For a matrix satisfying $R^T R = I$ to be specifically a rotation matrix (and not a reflection), it must also have a determinant of +1 ($\det(R) = +1$). However, the reference explicitly lists the 6 constraints related to the unit vector and orthogonality properties of the column vectors.
Here is a summary of the constraints:
| Constraint Type | Description | Number | Mathematical Representation (for 3x3 matrix $R=[\mathbf{c}_1 | \mathbf{c}_2 | \mathbf{c}_3]$) |
| :-------------- | :--------------------------------------------- | :----- | :--------------------------------------------------------------------------------------------- |
| Unit Vector | Each column vector has a norm of 1 | 3 | $|\mathbf{c}_1|=1, |\mathbf{c}_2|=1, |\mathbf{c}_3|=1$ |
| Orthogonality | Dot product of distinct column vectors is zero | 3 | $\mathbf{c}_1 \cdot \mathbf{c}_2=0, \mathbf{c}_1 \cdot \mathbf{c}_3=0, \mathbf{c}_2 \cdot \mathbf{c}_3=0$ |
