Orthogonal Transformation
Definition
$n \times n$ The matrix transformation $T_{A}: \mathbb{R}^{n} \to \mathbb{R}^{n}$ corresponding to the orthogonal matrix $A$ is called an orthogonal transformation.
$$ \mathbf{y} = T_{A}(\mathbf{x}) = A \mathbf{x} \quad (A^{\mathsf{T}}A = I) $$
Explanation
An orthogonal matrix and an orthogonal transformation share the same mathematical essence, but referring to it as a transformation rather than a matrix indicates a viewpoint more focused on the function. An orthogonal transformation also satisfies the same properties satisfied by orthogonal matrices.
Properties
(a) An orthogonal transformation is a linear transformation.
(b) An orthogonal transformation preserves the inner product (angle). $$ \Braket{T_{A}\mathbf{x} , T_{A}\mathbf{y}} = \braket{\mathbf{x}, \mathbf{y}} $$
(c) An orthogonal transformation preserves the norm (length). $$ \left\| T_{A} \mathbf{x} \right\| = \left\| \mathbf{x} \right\| $$