      IEVref: 102-03-18 ID: Language: en Status: Standard    Term: Hermitian product Synonym1:  Synonym2:  Synonym3:  Symbol: Definition: complex scalar, denoted by $U\cdot {V}^{*}$, attributed to any pair of vectors U and V in a complex vector space by a given function, with the following properties: $V\cdot {U}^{*}={\left(U\cdot {V}^{*}\right)}^{*}$, $\left(\alpha \text{\hspace{0.17em}}U\right)\cdot {V}^{*}=\alpha \text{\hspace{0.17em}}\left(U\cdot {V}^{*}\right)$ and $U\cdot {\left(\beta V\right)}^{*}={\beta }^{*}\left(U\cdot {V}^{*}\right)$ where α and β are complex scalars, $\left(U+V\right)\cdot {W}^{*}=U\cdot {W}^{*}+\text{\hspace{0.17em}}V\cdot {W}^{*}$ for every vector W existing in the same vector space, $U\cdot {U}^{*}>0$ for $U\ne 0$, where the asterisk denotes the conjugate vectorNote 1 to entry: 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: $U\cdot {V}^{*}=\sum _{i}{U}_{i}{V}_{i}{}^{*}$ Note 2 to entry: 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 to entry: The Hermitian product is denoted by a half-high dot (·) between the two symbols representing one vector and the conjugate of the other. Publication date: 2008-08 Source: Replaces: Internal notes: 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO 2017-08-24: replaced a ">" in 4th bullet point with a ">". LMO CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: