Orthogonal vectors and orthogonal matrices

A unit vector has norm 1: ‖x‖₂ = 1. Vectors x and y are orthogonal if xᵀy = 0; when both are nonzero, they meet at 90°. In ℝⁿ, at most n nonzero vectors can be mutually orthogonal. Vectors that are orthogonal and of unit length are orthonormal.

An orthogonal matrix is a square matrix whose rows are mutually orthonormal and whose columns are mutually orthonormal: AᵀA = AAᵀ = I. So its inverse is just its transpose, A⁻¹ = Aᵀ, which is very cheap to compute.

Pay attention to the name: the rows of an “orthogonal” matrix are not merely orthogonal but fully orthonormal. There is no special term for a matrix whose rows or columns are orthogonal but not orthonormal.