An orthogonal transformation of a real inner-product space is a linear map satisfying . It preserves inner products, lengths, angles, and therefore Gaussian curvature when applied to an embedded surface in Euclidean space.
Let be an arbitrary basis of a real inner product space, with Gram matrix . If a linear map has matrix in this basis, thenConsequently is an orthogonal transformation exactly when . The simpler orthogonal matrix condition applies directly when the chosen basis is an orthonormal basis, so that .
For example, take a two-dimensional vector space with weighted Euclidean inner product having . Thensatisfies , but . Thus it represents an orthogonal transformation even though it is not an orthogonal matrix.
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An orthogonal transformation is a linear transformation that preserves the inner product of vectors, which in turn means it also preserves lengths and angles between vectors. In practical terms, if you apply an orthogonal transformation to a set of vectors, the transformed vectors will maintain their geometric relationships. Mathematically, a transformation \( T: \mathbb{R}^n \to \mathbb{R}^n \) can be represented using a matrix \( A \).