nLab orthogonal matrix



Linear algebra

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Basic facts




An orthogonal matrix is a square matrix AA whose transpose matrix equals its inverse matrix A T=A 1A^T = A^{-1}, hence such that A TA=IdA^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.


Every real matrix AA can be factorized A=QRA = Q R where QQ is orthogonal and RR 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 (AA such that Id+A{\operatorname{Id}}+A is invertible) - a chart which is often having an advantage over using the exponential map.


Last revised on March 15, 2024 at 22:51:02. See the history of this page for a list of all contributions to it.