Sesquilinear form

From Exampleproblems

In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and conjugate-linear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half". Compare with a bilinear form, which is linear in both arguments.

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.

For a fixed z in V the map $w \mapsto \phi(z,w)$ is a linear functional on V (i.e. an element of the dual space V*). 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.