Science Fair Project Encyclopedia
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear 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.
Conventions differ as to which argument should be antilinear. We take the first to be antilinear 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
for all x,y,z,w ∈ V and all a ∈ C.
For a fixed z in V the map is a linear functional on V (i.e. an element of the dual space V*). Likewise, the map is an antilinear functional on V.
Given any sesquilinear form φ on V we can define a second sesquilinear form ψ via the conjugate transpose:
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.
A Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form h : V × V → C such that
The standard Hermitian form on Cn is given by
The components of H are given by Hij = h(ei, ej).
The quadratic form assoctiated to a Hermitian form
- Q(z) = h(z,z)
A skew-Hermitian form (also called a antisymmetric sesquilinear form), is a sesquilinear form ε : V × V → C such that
Every skew-Hermitian form can be written as i times a Hermitian form.
The quadratic form assoctiated to a skew-Hermitian form
- Q(z) = ε(z,z)
is always pure imaginary.
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details