Orthogonal matrix

A real matrix is orthogonal (or, more precisely, orthonormal) when it has an inverse equal to its transpose

P^T=P^-1

The term comes from the fact that the canonical orthonormal basis of the <math>\mathbb{R}^n\,</math> is transformed by any orthonormal matrix (and only by orthonormal matrices) into another orthonormal basis.

Each orthonormal matrix represents one orthonormal basis, and reciprocally:

• If the columns of an orthonormal matrix are taken to be a set of vectors, then this set is an orthonormal basis
• If an orthonormal basis is written as the columns of a matrix, then this matrix is orthonormal
• These properties are also valid for lines instead of columns

The concept generalizes to complex matrices as unitary matrices; however, in this case, instead of the transpose it's necessary to use the conjugate-transpose.