nLab
orthogonal matrix

Contents

Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Definition

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=1A^T A = 1 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 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 I+AI+A is invertible) - a chart which is often having an advantage over using the exponential map.

References

Last revised on February 5, 2021 at 00:48:52. See the history of this page for a list of all contributions to it.