• $V\cdot U^{*}={(U\cdot V^{*})}^{*}$,
• $(\alpha U)\cdot V^{*}=\alpha (U\cdot V^{*})$ and $U\cdot {(\beta V)}^{*}={\beta }^{*}(U\cdot V^{*})$ where α and β are complex scalars,
• $(U+V)\cdot W^{*}=U\cdot W^{*}+V\cdot W^{*}$ for every vector W existing in the same vector space,
• $U\cdot U^{*}>0$ for $U\ne 0$,

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 $U_i$of the vector U and the conjugate of the corresponding coordinate $V_i$ of the vector V:

$U\cdot V^{*}={\displaystyle \sum _i{U}_i{V}_i{}^{*}}$

NOTE 2 For two complex vectors or two complex vector quantities U and V either the Hermitian product $U\cdot V^{*}$ or a conjugate Hermitian product $U^{*}\cdot V$ may be used depending on the application. The Hermitian product $U\cdot U^{*}$ or $U^{*}\cdot U$ 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.