Orthogonaltransformation

An orthogonal transformation is a linear transformation of a vector space that preserves the dot product of vectors. In simpler terms, it's a transformation that doesn't change the lengths of vectors or the angles between them. This property means that orthogonal transformations preserve distances and orientations.

The most common examples of orthogonal transformations in Euclidean space are rotations and reflections. Rotations preserve the handedness of a coordinate system, while reflections reverse it. Mathematically, an orthogonal transformation can be represented by an orthogonal matrix. A matrix is orthogonal if its transpose is equal to its inverse, meaning that multiplying the matrix by its transpose results in the identity matrix.

Orthogonal transformations are fundamental in many areas of mathematics and physics. In linear algebra, they are used to simplify problems by changing the basis of a vector space to a more convenient one. In geometry, they are used to describe rigid motions. In computer graphics, rotations and reflections are essential for manipulating objects in 3D space. In quantum mechanics, orthogonal transformations are used to describe changes in the state of a quantum system. The preservation of dot products ensures that physical quantities that depend on these, such as energy, remain invariant under these transformations.