Paulimatrices

Paulimatrices are a class of square matrices defined over a field F that preserve a fixed nondegenerate bilinear form, encoded by a matrix J. The family includes orthogonal, pseudo-orthogonal, and symplectic matrices as special cases depending on the choice of J.

Let J be a fixed nondegenerate matrix in GL_n(F). A ∈ M_n(F) is a Paulimatrix with respect to J if A^T J A = J. The set P_J = {A ∈ GL_n(F) | A^T J A = J} is a Lie group known as the Pauligroup associated with J. Its Lie algebra consists of X with X^T J + J X = 0.

Examples and structure: If J = I_n, Paulimatrices are orthogonal matrices O(n). If J has signature (p, q), they form the pseudo-orthogonal group O(p, q). If J is the standard symplectic form, the Paulimatrix group is the symplectic group Sp(2m, F). Thus Paulimatrices unify several classical matrix groups under a single formalism.

Properties and applications: Paulimatrices are invertible, with A^{-1} = J^{-1} A^T J. The groups are Lie groups with familiar topological and algebraic properties. They appear in areas like physics, computer graphics, and numerical linear algebra, particularly where preservation of a bilinear form is required.

See also: Orthogonal matrix, Symplectic matrix, Bilinear form, Matrix group. The term “Paulimatrix” appears in a limited set of texts and is sometimes used as a generic label rather than a standard name.