'''Definition: In a finite dimensional space, given any linear operator *A* defined on the whole space, there exists an operator *A*^{ † } such that '''

⟨*ϕ*, *A**χ*⟩ = ⟨*A*^{ † }*ϕ*, *χ*⟩, *ϕ*, *χ* ∈ H.

** A^{ † } is uniquely determined and is called the adjoint of A.**

It must be noticed that for bounded operators the adjoint operator can be defined naturally, whereas for unbounded operators this can be done only if the domain of *A*, D(*A*), is dense in H and only for those vectors *ϕ* for which ⟨*ϕ*, *A**χ*⟩ is a continuous function of *χ*. The set of these vectors is a vectorial subset of H and by definition it will be the domain, D(*A*^{ † }), of *A*^{ † }.

**Definition:** A linear operator *A* is said to be:

- a
*Hermitian*operator if*A*⊆*A*^{ † }, i.e. if

⟨*ϕ*, *A**ϕ*⟩ = ⟨*A**ϕ*, *ϕ*⟩, *ϕ* ∈ D(*A*)

;

- a
*symmetric operator*if*A*⊆*A*^{ † }and $\; \overline{\mathcal{D}(A) }=\mathcal{H}$, where $\; \overline{\mathcal{D}(A) }$ is the complement of D(*A*); - a
**self-adjoint**operator if*A*=*A*^{ † }and $\; \overline{\mathcal{D}(A) }=\mathcal{H}$.

For bounded linear operators, and in particular for linear operators in finite dimensional Hilbert spaces, the three definitions coincide.

'''Proposition: A necessary and sufficient condition for the linear operator *A* to be *Hermitian* is that '''

⟨*ϕ*, *A**χ*⟩ = ⟨*A**ϕ*, *χ*⟩, *ϕ*, *χ* ∈ H.

**Proof:** This equality certainly implies the definition 1. of Hermitian operator. The converse is true because of the following identity, which can be easily verified:

4⟨*ϕ*, *A**χ*⟩ = ⟨*ϕ* + *χ*, *A*(*ϕ* + *χ*)⟩⟨*ϕ* − *χ*, *A*(*ϕ* − *χ*)⟩ − *i*⟨*ϕ* + *i**χ*, *A*(*ϕ* + *i**χ*)⟩ + *i*⟨*ϕ* − *i**χ*, *A*(*ϕ* − *i**χ*)⟩.

The matrix which represents the *adjoint* of a linear operator *A* in any orthonormal basis is the Hermitian conjugate of the matrix representing *A*. Indeed its matrix elements are given by:

*A*_{m, n}^{ * } = ⟨*ξ*_{m}, *A**ξ*_{n}⟩^{ * } = ⟨*A*^{ † }*ξ*_{m}, *ξ*_{n}⟩^{ * } = ⟨*ξ*_{n}, *A*^{ † }*ξ*_{m}⟩ = *A*_{n, m}^{ † }

.

So, if a bounded linear operator is **self-adjoint**, it is represented by a Hermitian matrix in any orthonormal basis.

Category:Linear Algebra category:Handbook of Quantum Information