Coordinate Transformation Matrix

Relevant parts to questions...

• Using $x_i^{\prime }=L_{ij}{x}_j$ (forward) and $x_i=L_{ji}{x}_j^{\prime }$ (inverse).
• Using $L_{ij}={\mathbf{e}}_i^{\prime }\cdot {\mathbf{e}}_j$ to read off entries from two bases.
• Using orthogonality $L_{ij}{L}_{kj}={\delta }_{ik}=L_{ji}{L}_{jk}$ to cancel pairs of $L$s into a Kronecker Delta.
• Using $|L|=1$ and $L^{-1}=L^T$ to sanity-check $L$.

The Coordinate Transformation Matrix (or rotation matrix) $L$ encodes a rigid rotation between two orthonormal bases $\{{\mathbf{e}}_j\}$ and $\{{\mathbf{e}}_i^{\prime }\}$. Its entries are defined as the dot products of new and old basis vectors:

$$\boxed{L_{ij}={\mathbf{e}}_i^{\prime }\cdot {\mathbf{e}}_j}$$

Equivalently, $L_{ij}$ is the cosine of the angle between ${\mathbf{e}}_i^{\prime }$ and ${\mathbf{e}}_j$. Because every coordinate is just a dot product of the position vector with a basis vector ($x_i^{\prime }={\mathbf{e}}_i^{\prime }\cdot \mathbf{r}$), the same matrix transforms coordinates:

$$\boxed{x_i^{\prime }=L_{ij}{x}_j}$$

In 2D, rotating the axes by an angle $\theta$ gives the familiar form:

$$L=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$$

Properties

Because $L$ represents a rotation, it has the following tightly-linked properties:

• Orthogonal::$L^T=L^{-1}$, so $L{L}^T=L^{T}L=I$; in suffix form, $L_{ij}{L}_{kj}={\delta }_{ik}=L_{ji}{L}_{jk}$.
• Unit determinant::$|L|=1$ (a rigid rotation neither scales nor reflects).
• Inverse is a rotation by $-\theta$::the new-to-old transform is just the transpose, i.e., $x_i=L_{ji}{x}_j^{\prime }$.
• Each ${\mathbf{e}}_i^{\prime }$ expands as ${\mathbf{e}}_i^{\prime }=L_{ij}{\mathbf{e}}_j$::the rows of $L$ are the coefficients of the new basis in terms of the old.

Applications

1. Transforming coordinates, using $x_i^{\prime }=L_{ij}{x}_j$ forward and $x_i=L_{ji}{x}_j^{\prime }$ backward.
2. Transforming vectors, via $v_i^{\prime }=L_{ij}{v}_j$ (the defining property of a vector).
3. Transforming higher-rank objects, with one factor of $L$ per free index; e.g., $T_{ij}^{\prime }=L_{im}{L}_{jn}{T}_{mn}$ for a rank-2 Tensor Transformation Rule.
4. Deriving the Jacobian::$\frac{\partial x_i^{\prime }}{\partial x_j}=L_{ij}$ and $\frac{\partial x_i}{\partial x_j^{\prime }}=L_{ji}$, which bridges index calculus and rotations.

Example

Given the orthonormal basis $\{\mathbf{i},\mathbf{j},\mathbf{k}\}$ and a new basis ${\mathbf{e}}_1^{\prime }=\mathbf{i}$, ${\mathbf{e}}_2^{\prime }=\mathbf{j}$, ${\mathbf{e}}_3^{\prime }=\mathbf{k}$ rotated about $\mathbf{k}$ by $\theta$, read off $L_{ij}={\mathbf{e}}_i^{\prime }\cdot {\mathbf{e}}_j$ to get $L=\left(\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ -\sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right)$. The point $(1,0,0)$ transforms to $x_i^{\prime }=L_{ij}{x}_j=\boxed{(\cos \theta ,-\sin \theta ,0)}$.

Rotation matrix in, tensor transformation out.