- Hermitian Adjoint of A

: Mathematical definition and properties of the Hermitian Adjoint operator

1. Definition of $L : V \to V^*$

$L : V \to V^*$, $v \mapsto \tilde{v}$, where $L(v)(w) = \tilde{v}(w) = (v \mid w)$

It’s important to note that $L$ is a conjugate-linear map.

$A^\dagger$ represents the Hermitian adjoint matrix of $A$ and is defined as follows.

$$A^\dagger : V \to V, \quad A^\dagger = L^{-1} \circ A^T \circ L$$

Therefore, $A^\dagger(v) = L^{-1}(A^T(\tilde{v}))$. Since $A^\dagger(v) = A^\dagger v$ and $A^T(\tilde{v}) = A^T\tilde{v}$, we have

$$L(A^\dagger v) = A^T\tilde{v}$$

Rewriting this,

$$L(A^\dagger v)(w) = (A^T\tilde{v})(w), \quad w \in V$$

Using the relation $(A^T\tilde{v})(w) = \tilde{v}(Aw)$, this is equivalent to:

$$( A^\dagger v | w ) = \tilde{v}(Aw) = (v | Aw),$$

$$( A^\dagger v | w ) = (v | Aw)$$

This relationship is commonly used as the definition of $A^\dagger$.

When $v = e_i$ and $w = e_j$:

$$(\sub{A^\dagger)}{i}^k e_k | e_j) = (\sub{A^\dagger)}{i}^k )^* \delta_{jk} = (\sub{A^\dagger)}{i}^j)^* = A^{\dagger j*}_{\ i}$$

Also,

$$(e_i | A_j^k e_k) = A_j^k \delta_{ik} = A_j^i$$

Therefore,

$$\sub{A^{\dagger j*}}{i} = \sub{A}{j}^i \quad \text{or} \quad \sub{A^{j*}}{i} = \sub{A^{\dagger i}}{j}$$