Sesquilinear Form - Definition and Conventions

Definition and Conventions

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 convention used by essentially all physicists and originates in Dirac's bra-ket notation in quantum mechanics. The opposite convention is perhaps more common in mathematics but is not universal.

Specifically a map φ : V × VC is sesquilinear if

\begin{align}
&\phi(x + y, z + w) = \phi(x, z) + \phi(x, w) + \phi(y, z) + \phi(y, w)\\
&\phi(a x, b y) = \bar a b\,\phi(x,y)\end{align}

for all x,y,z,wV and all a, bC.

A sesquilinear form can also be viewed as a complex bilinear map

where is the complex conjugate vector space to V. By the universal property of tensor products these are in one-to-one correspondence with (complex) linear maps

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 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:

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.

Read more about this topic:  Sesquilinear Form

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