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IEVref: | 102-03-18 | ID: | |

Language: | en | Status: Standard | |

Term: | Hermitian product | ||

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Definition: | complex scalar, denoted by $U\cdot {V}^{*}$, attributed to any pair of vectors and U in a complex vector space by a given function, with the following properties: V- $V\cdot {U}^{*}={(U\cdot {V}^{*})}^{*}$,
- $(\alpha \text{\hspace{0.17em}}U)\cdot {V}^{*}=\alpha \text{\hspace{0.17em}}(U\cdot {V}^{*})$ and $U\cdot {(\beta V)}^{*}={\beta}^{*}(U\cdot {V}^{*})$ where
*α*and*β*are complex scalars, - $(U+V)\cdot {W}^{*}=U\cdot {W}^{*}+\text{\hspace{0.17em}}V\cdot {W}^{*}$ for every vector
existing in the same vector space,*W* - $U\cdot {U}^{*}>0$ for $U\ne 0$,
where the asterisk denotes the conjugate vector Note 1 to entry: In an is the sum of the products of each coordinate ${U}_{i}$of the vector V and the conjugate of the corresponding coordinate ${V}_{i}$ of the vector U: V$U\cdot {V}^{*}={\displaystyle \sum _{i}{U}_{i}{V}_{i}{}^{*}}$ Note 2 to entry: For two complex vectors or two complex vector quantities 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. VNote 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 | ||

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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 | ||

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- $V\cdot {U}^{*}={(U\cdot {V}^{*})}^{*}$,
- $(\alpha \text{\hspace{0.17em}}U)\cdot {V}^{*}=\alpha \text{\hspace{0.17em}}(U\cdot {V}^{*})$ and $U\cdot {(\beta V)}^{*}={\beta}^{*}(U\cdot {V}^{*})$ where
*α*and*β*are complex scalars, - $(U+V)\cdot {W}^{*}=U\cdot {W}^{*}+\text{\hspace{0.17em}}V\cdot {W}^{*}$ for every vector
existing in the same vector space,*W* - $U\cdot {U}^{*}>0$ for $U\ne 0$,

where the asterisk denotes the conjugate vector

Note 1 to entry: In an *n*-dimensional space with orthonormal base vectors the Hermitian product of two vectors ** U** and

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

Note 2 to entry: For two complex vectors or two complex vector quantities ** U** and

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.