For Inner Product spaces V,W and a linear map α between the two, the adjoint of α is a linear map α* satisfying

<α(v), w> = <v,α*(w)> for all v ∈ V and w ∈ W. where < , > denotes the inner product (notation varies).

Whilst linear maps are equivalent to matrices, the notation has become slightly complex for adjoints. Firstly, there is a source of confusion with the adjoint matrix. Briefly, this, denoted adj(A) and often described instead as the adjugate matrix of A, is a matrix derived from A in terms of determinants of sub-matrices. It is *not* the matrix representation of the adjoint map.

In line with the notation for linear maps, A* may be used to indicate the matrix representing the adjoint of the map represented by A. However, this notation is often also used to denote complex conjugation of a matrix. Since the concept of an adjoint requires conjugation, and HTML can't supply the usual bar notation, it makes sense to reserve * for that operation and use the alternative notation of A^{H} for the adjoint. In fact, this indicates the hermitian conjugate of A (that is, in our notation, A*^{T}), but this is appropriate since, for a finite dimensional vector space, the adjoint of a map α represented by A in an orthonormal basis is represented by precisely A^{H}.

Whilst the finite dimensional case will always yield an adjoint (via the map represented by matrix A^{H}), it is not always the case that an adjoint can be constructed for an infinite dimensional space. However, if the adjoint does exist, it is unique:

**Proof:**Returning to maps instead of matrices, consider α with adjoint α* and a rival adjoint α'. Then by the adjoint defintion:

< v,α'(w) > = < α(v),w > = < v,α*(w) > ∀v,w

Then < v,α'(w)-α*(w) > =0 ∀v,w

So by Inner Product defintion, α'(w)-α*(w)=0 ∀w

So α' = α*

Any claimed rival α' for the adjoint is in fact the original adjoint α*, so the adjoint map is unique.

By simple manipulation of the axioms for inner products and the adjoint definition, the following properties hold:

Let α:V→W, β:W→Z, γ:V→W all have corresponding adjoints α*, β*, γ*. Then

- (α + γ)* = α* + γ*
- (λα)* = λ*α* (Where λ* is the complex conjugate of λ a constant from the field)
- (βoα)* = α*oβ*
- (α*)* = α

Of particular interest are so called self-adjoint maps: those which are their own adjoint. In terms of matrices, this makes them their own hermitian conjugate, and hence by definition a hermitian matrix (for the real case, this simply means it is symmetric). Such maps are useful because they allow the spectral theorem to be applied: the eigenvectors of such a map will constitute an orthonormal basis for the vector space, with all the corresponding eigenvalues being real. This allows for diagonalisation.