| complex scalar, denoted by , attributed to any pair of vectors U and V in a complex vector space by a given function, with the following properties: |
- and where α and β are complex scalars,
- for every vector W existing in the same vector space,
- for ,
where the asterisk denotes the conjugate vector
NOTE 1 In an n-dimensional space with orthonormal base vectors the Hermitian product of two vectors U and V is the sum of the products of each coordinate of the vector U and the conjugate of the corresponding coordinate of the vector V:
NOTE 2 For two complex vectors or two complex vector quantities U and V either the Hermitian product or a conjugate Hermitian product may be used depending on the application. The Hermitian product or is a real scalar or a real scalar quantity, respectively.
NOTE 3 The Hermitian product is denoted by a half-high dot (·) between the two symbols representing one vector and the conjugate of the other.