Adjoint

Definition

1. $T$ is a Linear Operator on Inner Product Space $V$
2. Then, $T$ has an adjoint $T^{\ast }$ on $V$ such that $⟨T\alpha |\beta ⟩=⟨\alpha |T^{\ast }\beta ⟩,\forall \alpha ,\beta \in V$ For matrixes, the adjoint is the Conjugate Transpose

Properties

1. If an adjoint exists, it is Unique
2. The adjoint depends on $T$ and the Standard Inner Product of the Inner Product Space
3. If $V$ is finite, an adjoint $T^{\ast }$ always exists

Theorems

• Adjoints Exist for Finite Dimensional Space
• Matrix Representation of Adjoints with Respect to Orthonormal Basis
• Self Adjoint