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

Famous quotes containing the words definition and/or conventions:

    The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.
    William James (1842–1910)

    Art, it seems to me, should simplify. That, indeed, is very nearly the whole of the higher artistic process; finding what conventions of form and what detail one can do without and yet preserve the spirit of the whole—so that all that one has suppressed and cut away is there to the reader’s consciousness as much as if it were in type on the page.
    Willa Cather (1873–1947)