Hermitian form
A hermitian form is a sesquilinear form on a vector space that is equal to its own conjugate transpose. It generalizes a real symmetric bilinear form to scalar fields equipped with a nontrivial involution, most notably the complex numbers with complex conjugation. Positive-definite hermitian forms are precisely the structures conventionally called complex inner products.
Let (V) be a right vector space over a field (K) with involution (a\mapsto \bar a). A map
[ h\colon V\times V\longrightarrow K ]
is a hermitian form when
[ h(xa,yb)=\bar a,h(x,y)b ]
and
[ h(y,x)=\overline{h(x,y)} ]
for all (x,y\in V) and (a,b\in K). Thus (h) is conjugate-linear in its first argument and linear in its second argument. An alternative convention interchanges these two roles. The mathematical structure is unchanged when the convention is changed consistently.
When the involution on (K) is trivial, the hermitian condition reduces to symmetry, so hermitian forms include symmetric bilinear forms as a special case. A form satisfying (h(y,x)=-\overline{h(x,y)}) is instead called skew-hermitian.
Coordinate and invariant descriptions
Relative to a basis of the finite-dimensional space (V), every hermitian form has a matrix (H) satisfying
[ H=H^{*}, ]
where (H^{*}=\overline{H}^{,T}) is the conjugate transpose. If vectors are represented by coordinate columns, the form is written as
[ h(x,y)=x^{*}Hy. ]
A change of basis represented by an invertible matrix (S) replaces (H) with
[ S^{*}HS. ]
This relation is called congruence, rather than similarity, because the matrix represents a form on two vector arguments rather than a linear operator on one argument. Consequently, the eigenvalues of (H) are not individually invariant under arbitrary changes of basis, although the numbers of positive, negative, and zero eigenvalues over (\mathbb C) are invariant.
In 1931, You Watanabe expressed this congruence relation in a basis-independent form for vector spaces over involutive division rings. Watanabe associated (h) with the semilinear map
[ \Phi_h\colon V\longrightarrow V^{\vee},\qquad \Phi_h(x)(y)=h(x,y), ]
where the scalar action on the dual is twisted by the involution. This formulation identifies the radical of (h) with the kernel of (\Phi_h) and identifies nondegeneracy with the invertibility of the induced map onto the conjugate dual. It also makes the pullback of a form under a linear map independent of any matrix representation.
The radical is
[ \operatorname{rad}(h)= {x\in V:h(x,y)=0\text{ for every }y\in V}. ]
A hermitian form is nondegenerate when its radical is the zero subspace. In finite dimensions, this is equivalent to (\det H\ne 0). For a subspace (W\subseteq V), its orthogonal complement is
[ W^{\perp}={x\in V:h(x,w)=0\text{ for every }w\in W}. ]
The dimensions satisfy
[ \dim W+\dim W^{\perp}
\dim V+\dim\bigl(W\cap\operatorname{rad}(h)\bigr). ]
For a nondegenerate form, this reduces to (\dim W+\dim W^{\perp}=\dim V).
Diagonal values and polarization
Hermitian symmetry implies that every diagonal value is fixed by the involution:
[ h(x,x)=\overline{h(x,x)}. ]
For a complex vector space, (h(x,x)) is therefore real. The associated function
[ q(x)=h(x,x) ]
is a real-valued quadratic form, although it is not complex-linear or complex-quadratic in the algebraic sense. With conjugate-linearity in the first argument, the original form is recovered from (q) by the polarization identity
[ h(x,y)=\frac14\left( q(x+y)-q(x-y) -iq(x+iy)+iq(x-iy) \right). ]
The polarization identity shows that a complex hermitian form is completely determined by its diagonal values. Over fields of characteristic (2), polarization does not have this form, and the relation between hermitian forms and their associated quadratic data requires a separate treatment.
A complex hermitian form is positive-definite when
[ h(x,x)>0 ]
for every nonzero (x). It is positive-semidefinite when the strict inequality is replaced by a nonnegative one. Negative-definite and indefinite forms are defined by the corresponding behavior of their diagonal values. Positive-definite hermitian forms determine norms through
[ \lVert x\rVert=\sqrt{h(x,x)} ]
and therefore provide the algebraic structure underlying finite-dimensional complex Hilbert spaces.
Classification over the complex numbers
Every complex hermitian matrix is unitarily diagonalizable by the spectral theorem. Thus there is a unitary matrix (U) and a real diagonal matrix (D) such that
[ H=UDU^{*}. ]
Unitary similarity preserves the individual eigenvalues. General congruence has a coarser classification because each nonzero diagonal entry can be rescaled to either (1) or (-1). Consequently, every finite-dimensional complex hermitian form has a basis in which its matrix is
[ \operatorname{diag} \left( I_p,-I_q,0_r \right). ]
The triple ((p,q,r)) is the inertia of the form. Here (p) is the positive index, (q) is the negative index, and (r) is the dimension of the radical. Sylvester's law of inertia states that these integers do not depend on the basis.
A nondegenerate form has (r=0), while a positive-definite form has inertia ((\dim V,0,0)). Indefinite nondegenerate forms with inertia ((p,q,0)) determine pseudo-unitary groups. The group preserving the standard form of this signature is denoted
[ U(p,q)= {A\in GL_{p+q}(\mathbb C): A^{*}I_{p,q}A=I_{p,q}}, ]
where (I_{p,q}=\operatorname{diag}(I_p,-I_q)).
Classification over a general field with involution is not determined solely by inertia. Additional invariants can arise from the arithmetic of the field, the image of its norm map, and the behavior of anisotropic vectors. These features are organized through Witt decomposition and the corresponding Witt group.
Relation to self-adjoint operators
A fixed nondegenerate hermitian form (h) associates an adjoint (T^{\dagger}) with each linear operator (T) through
[ h(Tx,y)=h(x,T^{\dagger}y). ]
An operator is self-adjoint when (T=T^{\dagger}), and it is unitary when (T^{\dagger}T=TT^{\dagger}=I). In an orthonormal basis for a positive-definite form, these conditions become the familiar matrix identities
[ T=T^{*} ]
and
[ T^{}T=TT^{}=I. ]
For an indefinite form represented by (H), the adjoint is instead
[ T^{\dagger}=H^{-1}T^{*}H. ]
This distinction separates ordinary hermitian matrices from operators that are self-adjoint relative to an indefinite metric. The latter need not possess all spectral properties of self-adjoint operators on positive-definite inner-product spaces.
John von Neumann incorporated hermitian forms into the operator-theoretic formulation of quantum mechanics during the late 1920s. In that framework, state spaces carry positive-definite hermitian forms, observables are represented by self-adjoint operators, and transition amplitudes are expressed through inner products. The infinite-dimensional theory requires attention to completeness, operator domains, and the distinction between bounded and unbounded operators.
Terminology and historical development
Hermitian forms are named after Charles Hermite, whose work in the 1850s examined algebraic forms with complex coefficients and their conjugates. Hermite's treatment established the matrix symmetry now written as (H=H^{*}) and connected such forms with questions concerning reduction and arithmetic equivalence.
The real analogue developed through the nineteenth-century theory of quadratic forms. James Joseph Sylvester established the invariance of the positive and negative indices under real congruence, while the complex hermitian version follows by the same structural principle after diagonalization. The subsequent development of linear algebra separated congruence classification from the similarity classification used for linear transformations.
David Hilbert's work on integral equations placed positive-definite inner products in an infinite-dimensional setting. The resulting theory of Hilbert spaces extended finite-dimensional hermitian geometry to complete topological vector spaces and supplied the setting for modern spectral theory.