Adjoint Operator

linear-algebra

Definition

Adjoint Operator

Let $T:\mathcal{H}\to \mathcal{H}$ be a linear operator on a complex inner product space $\mathcal{H}$. The adjoint of $T$ is the unique operator $T^{†}$ satisfying

$$⟨Tx,y⟩=⟨x,T^{†}y⟩$$

for all $x,y\in \mathcal{H}$.

Example: For

$$A=\left(\begin{array}{cc}1& i\\ 2& 3\end{array}\right),$$

one has

$$A^{†}=\left(\begin{array}{cc}1& 2\\ -i& 3\end{array}\right).$$

Intuition

The adjoint $T^{†}$ is the operator obtained when one moves $T$ from the first slot of the inner product to the second slot. Thus

$$⟨Tx,y⟩=⟨x,T^{†}y⟩$$

means that applying $T$ to $x$ has exactly the same effect on the inner product as applying $T^{†}$ to $y$.

Over the real numbers, the adjoint is just the transpose. Hence $T^{†}$ is the complex analogue of “flipping” a matrix so that it acts on the other side of a dot product.

In complex spaces, the inner product uses complex conjugation. Therefore merely transposing a matrix is not enough; one must also conjugate its entries. This is why the adjoint is the conjugate transpose rather than only the transpose.

Matrix

If $T$ is represented by a matrix $A$, then the adjoint is given by the conjugate transpose

$$A^{†}={\overline{A}}^T.$$