adjungierten

Adjungierten is a term used in German mathematics to refer to two related notions in linear algebra: the adjoint of a linear operator and, in older or more technical texts, the adjugate (classical adjoint) matrix. The exact meaning depends on context, but both ideas concern a form of duality with respect to an inner product or a determinant.

In the context of operators on inner product spaces, the adjoint (Adjungierte) of a linear map A is the unique map A* that satisfies ⟨Ax, y⟩ = ⟨x, A*y⟩ for all vectors x and y. If the space is finite dimensional and one uses the standard inner product, the matrix of A* is the conjugate transpose of A, denoted A^H (or A†). In real spaces, A* coincides with the transpose A^T. The adjoint plays a central role in many results, such as the spectral theorem for Hermitian (A* = A) and unitary (A*A = AA* = I) operators. The adjoint of a product obeys (AB)* = B*A*, and the trace and determinant interact with adjoints via properties like det(A*) = overline(det(A)).

The adjugate (or adjoint in older terminology) is the transpose of the cofactor matrix. It satisfies A adj(A) = adj(A) A = det(A) I and provides an explicit formula for the inverse when det(A) ≠ 0: A^{-1} = adj(A)/det(A). In German texts, the phrase Adjungierte Matrix is sometimes used for this object, though modern usage often distinguishes it from the operator adjoint.

Applications of adjungierten concepts include solving linear systems, expressing orthogonality relations, and formulating the spectral theorem in complex inner product spaces. See also Hermitian transpose, inner product, and determinant.