Sesquilinear Form

N.B. Many authors, especially when working solely in a complex setting, refer to sesquilinear forms as bilinear forms.

Conventions differ as to which argument should be linear. We take the first to be conjugate-linear and the second to be linear. This is the physicist's convention — originating in Dirac's Bra-ket Notation in Quantum Mechanics — but is becoming more popular among mathematicians as well.

Specifically a map φ : ''V'' × ''V'' → C is sesquilinear if
:

\phi(x + y, z + w) = \phi(x, z) + \phi(x, w) + \phi(y, z) + \phi(y, w)\,

\phi(a x, y) = \bar{a}\,\phi(x,y)

\phi(x, ay) = a\,\phi(x,y)

for all ''x,y,z,w'' ∈ ''V'' and all ''a'' ∈ C.

). Likewise, the map $w \mapsto \phi(w,z)$ is a Conjugate-linear Functional on ''V''.

Given any sesquilinear form φ on ''V'' we can define a second sesquilinear form ψ via the Conjugate Transpose :
:

\psi(w,z) = \overline{\phi(z,w)}

In general, ψ and φ will be different. If they are the same then φ is said to be ''Hermitian''. If they are negatives of one another, then φ is said to be ''skew-Hermitian''. Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form.

HERMITIAN FORM

The term Hermitian form may also refer to a different concept than that explained below: it may refer to a certain Differential Form on a Hermitian Manifold .

A Hermitian form (also called a '''symmetric sesquilinear form'''), is a sesquilinear form ''h'' : ''V'' × ''V'' → '''C''' such that
:

h(w,z) = \overline{h(z, w)}

The standard Hermitian form on C^''n'' is given by
:

\langle w,z 
angle = \sum_{i=1}^n\overline{w}_i z_i

More generally, the Inner Product on any Hilbert Space is a Hermitian form.

If ''V'' is a finite-dimensional space, then relative to any Basis {''e''_''i''} of ''V'', a Hermitian form is represented by a Hermitian Matrix H:
:

h(w,z) = \overline{\mathbf{w}}^T \mathbf{Hz}

The components of H are given by ''H''_''ij'' = ''h''(''e''_''i'', ''e''_''j'').

The Quadratic Form associated to a Hermitian form

Q

is always Real . Actually one can show that a sesquilinear form is Hermitian Iff the associated quadratic form is real for all ''z'' ∈ ''V''.

SKEW-HERMITIAN FORM

A skew-Hermitian form (also called a '''antisymmetric sesquilinear form'''), is a sesquilinear form ε : ''V'' × ''V'' → '''C''' such that
:

arepsilon(w,z) = -\overline{arepsilon(z, w)}

Every skew-Hermitian form can be written as ''i'' times a Hermitian form.

If ''V'' is a finite-dimensional space, then relative to any Basis {''e''_''i''} of ''V'', a skew-Hermitian form is represented by a Skew-Hermitian Matrix A:
:

arepsilon(w,z) = \overline{\mathbf{w}}^T \mathbf{Az}

The quadratic form associated to a skew-Hermitian form

Q

is always pure Imaginary .

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Sesquilinear Form