endbmatrices

Endbmatrices are a class of matrices associated with endomorphisms of a finite-dimensional vector space V over a field F equipped with a fixed nondegenerate bilinear form B. The form B is represented by a matrix G in a chosen basis. For an endomorphism φ with matrix M, the B-adjoint φ† is defined by B(φ(u), v) = B(u, φ†(v)) for all u, v in V, and its matrix is M† = G^{-1} M^T G. An endbmatrix can be understood as the matrix representation of φ in this B-structured setting; properties of φ translate into relations between M and M†.

If a matrix satisfies M† = M, φ is B-self-adjoint; if M† = -M, φ is B-skew-adjoint. The collection of endbmatrices corresponding to endomorphisms forms an algebra, and classification is often done up to change of basis that preserves B (the B-orthogonal or B-symplectic group).

For a symmetric B (such as the standard dot product where G = I), endbmatrices reduce to familiar adjoint relations: M† = M^T and B-self-adjoint means M is symmetric.

Canonical forms: Under B-preserving similarity, endbmatrices admit structured decompositions that reveal invariant subspaces, analogous to the spectral theorem in the self-adjoint case.

Applications: Endbmatrices appear in representation theory, linear differential equations with conserved bilinear forms, and numerical linear algebra algorithms that respect a given bilinear form. See also Bilinear form, Adjoint, Orthogonal group, Symplectic group, Canonical form.