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Hermitian Form


A Hermitian form on a vector space V over the complex field C is a function f:V×V->C such that for all u,v,w in V and all a,b in R,

1. f(au+bv,w)=af(u,w)+bf(v,w).

2. f(u,v)=f(v,u)^_.

Here, the bar indicates the complex conjugate. It follows that

 f(u,av+bw)=a^_f(u,v)+b^_f(u,w),
(1)

which can be expressed by saying that f is antilinear on the second coordinate. Moreover, for all v in V, f(v,v)=f(v,v)^_, which means that f(v,v) in R.

An example is the dot product of C^n, defined as

 (u_1,...,u_n)·(v_1,...,v_n)=sum_(i=1)^nu_iv_j^_.
(2)

Every Hermitian form on C^n is associated with an n×n Hermitian matrix A such that

 f(X,Y)=XAY^_^(T),
(3)

for all row vectors X and Y of C^n. The matrix associated with the dot product is the n×n identity matrix.

More generally, if V is a vector space on a field K, and phi:K->K is an automorphism such that phi!=id_K, and phi^2=id_K, the notation phi(a)=a^_ can be used and a Hermitian form f on V can be defined by means of the properties (1) and (2).


See also

Hermitian Matrix

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Hermitian Form." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HermitianForm.html

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