nLab orthogonal matrix

Definition

An orthogonal matrix is a square matrix $A$ whose transpose matrix equals its inverse matrix $A^T = A^{-1}$, hence such that $A^T A = \operatorname{Id}$ under matrix multiplication.

Orthogonal matrices form a subgroup of the general linear group, namely the orthogonal group.

For a generalization see J-orthogonal matrix.

Properties

Every real matrix $A$ can be factorized $A = Q R$ where $Q$ is orthogonal and $R$ is a (say, upper) triangular matrix (wikipedia/QR decomposition which is a special case of Iwasawa decomposition for semisimple Lie groups). This is consequence of Gram-Schmidt orthogonalization. Similarly, every complex matrix can be factorized into a unitary and a complex upper triangular (complex) matrix.

Over the reals, the Cayley transform is a diffeomorphism between the linear space skewsymmetric matrices and an open subset of the Lie group of orthogonal matrices ($A$ such that ${\operatorname{Id}}+A$ is invertible) - a chart which is often having an advantage over using the exponential map.

Related concepts

• J-orthogonal matrix

• orthogonal group, Gram-Schmidt orthogonalization

References

• wikipedia/orthogonal matrix