The term adjoint is used for several constructions. The adjoint operator of an operator between Hilbert spaces is defined using their inner products. The analogous construction for Banach spaces is the Banach adjoint, while the construction obtained from integration by parts for a differential operator is its formal adjoint.
For a matrix, the phrase adjoint matrix is ambiguous. It can mean either the conjugate transpose or the adjugate matrix, which is also called the classical adjoint. In this work, conjugate transpose and adjugate matrix are used as the explicit names for these operations.
Other uses of the word include the adjoint curve in algebraic geometry, the adjoint representation of a Lie algebra, and an old name for the line graph of a graph.
