Sesquilinear Form
On a complex vector space
, a sesquilinear form is a map that is linear in the first argument and antilinear in the second argument, i.e. for all and , there holds Thus it can also be viewed as a bilinear map .
Conjugate of a Sesquilinear Form
Suppose
is a sesquilinear map, then we can define its conjugate by If , then it is called Hermitian; If , then it is called skew-Hermitian.
Proposition
Every sesquilinear form can be uniquely decomposed into a sum of a Hermitian form and a skew-Hermitian form.