The Adjoint

Adjoint Definition¶

Let $V$ be an inner product space defined over $\mathbb{C}$, with inner product $(\cdot, \cdot): \mathbb{C}^{n} \otimes \mathbb{C}^{n} \rightarrow \mathbb{C}^{1}$, and $L: V \rightarrow V$ a linear operator.

The adjoint operator of the linear operator $L$, denoted by $L^{\dagger}$, is the operator that satisfies the following relation

where $\bm{x}, \bm{y}~\in~V$.

Associated Matrix of the Adjoint¶

Let $M^{\mathfrak{B}}_{L}$ be the matrix associated to the linear map $L$ with respect to the orthonormal basis $\mathfrak{B}$ and the element in the $i$th row and $j$th colunm be $a_{ij}$.

The matrix associated to the linear map $L^{\dagger}$ with respect to the orthonormal basis $\mathfrak{B}$, $M_{L^{\dagger}}^{\mathfrak{B}}$, is given by taking the complex conjugate of all elements of $M_{L}^{\mathfrak{B}}$ and then transposing. Hence, the element in the $i$th row and $j$th column of $M_{L^{\dagger}}^{\mathfrak{B}}$ will be $a^{*}_{ji}$.

For this reason, the adjoint is also sometimes referred to as the conjugate transpose operations.

Properties of the Adjoint¶

Let $V, (\cdot, \cdot)$ be an inner product space defined over $\mathbb{C}$, let $L, T: V \rightarrow V$ be linear operators and $\lambda~\in~\mathbb{C}^{1}$.

• $(L+T)^{\dagger} = L^{\dagger} + T^{\dagger}$.
• $(\lambda L)^{\dagger} = \lambda^{*} L^{\dagger}$, where $(\cdot)^{*}$ is the complex conjugate.
• $(LT)^{\dagger} = T^{\dagger}L^{\dagger}.$
• $(L^{\dagger})^{\dagger} = L$.
• If $L^{-1}$ exists, then $(L^{\dagger})^{-1} = (L^{-1})^{\dagger}$.

Matrix Properties From the Adjoint¶