nLab
J-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

Given a square matrix JJ (representing a bilinear form) over a commutative ring kk, a JJ-orthogonal matrix is any square matrix AA of the same size over kk such that

A TJA=J A^T J A = J

where A TA^T is the transpose matrix of AA.

Most often one considers a case when JJ is diagonal. For J=1J = 1 one recovers the notion of an orthogonal matrix.

All J-orthogonal matrices over the ground field kk form an algebraic group denoted SO J(n,k)SO_J(n, k) or SO(n,k;J)SO(n,k; J) or alike; in real or complex case a Lie group. The matrices in the Lie algebra are J-skew-symmetric, A TJ+JA=0A^T J + J A = 0.

References

  • M M Postnikov, Lectures on geometry, Semester V, Lie groups and Lie algebras

  • Nicholas J. Higham, J-orthogonal matrices: properties and generation, doi

Last revised on February 5, 2021 at 01:59:44. See the history of this page for a list of all contributions to it.