An 直交変換 is a special type of 線形変換 in which the transformation preserves the 内積 of vectors, meaning that angles and lengths are maintained. In mathematical terms, if A is an orthogonal transformation represented by a matrix, then the columns of A form an orthonormal set of vectors. This can be expressed with the equation ATA = I, where AT denotes the transpose of A and I is the 単位行列.
正交変換は、一般的にさまざまな分野で使用されます コンピュータグラフィックス, robotics, and physics. In 3Dグラフィックス, for example, orthogonal transformations include rotations and reflections that allow for the manipulation of objects without altering their shape or size. This is crucial for maintaining visual fidelity when rendering scenes or modeling complex shapes.
In the context of coordinate systems, an orthogonal transformation can be used to change from one coordinate system to another in a way that preserves the geometric properties of the original shape. This makes them incredibly useful in データ処理 and analysis where maintaining the relationships between data points is important.
Furthermore, in the realm of machine learning, orthogonal transformations can assist in feature extraction and 次元削減, helping to retain meaningful information while reducing data complexity.