orthogonal matrix

A square matrix Q is orthogonal if its columns form an orthonormal set, equivalently Q^{\mathsf T}Q=I, which forces Q^{-1}=Q^{\mathsf T} — inverting costs nothing beyond transposing. Multiplying by Q never changes a vector's length or the angle between two vectors: an orthogonal matrix is precisely a rigid rotation or reflection fixed at the origin.