nLab J-orthogonal matrix

Definition

Given a square matrix $J$ (representing a bilinear form) over a commutative ring $k$, a $J$-orthogonal matrix is any square matrix $A$ of the same size over $k$ such that

$$A^T J A = J$$

where $A^T$ is the transpose matrix of $A$.

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

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

Related concepts

• orthogonal matrix

References

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

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