Adjoint

Two linear operators $A$ and $B$ are called adjoint when $$ \langle \eta,A \zeta\rangle = \langle B\eta,\zeta\rangle $$ for any two inputs $\zeta$ and $\eta$. Here $\langle \eta , \zeta \rangle$ is their inner product.

In a purely algebraic sense, $A$ and $B$ are defined on two different spaces:

• $A:V\rightarrow W$,
• $B:W^*\rightarrow V^*$,

where the star indicates the dual space. Then the above equation is understood via evaluation: $$ \eta(A \zeta) = (B\eta)(\zeta).$$ The operators are also called dual of each other: $$B=A^*.$$

In the case of finite dimensional spaces, their matrices are transpose of each other.