(Untitled) | (Untitled) | (Untitled) | (Untitled) | (Untitled) | Examples |

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

Language: | en | Status: backup | |

Term: | Hermitian product | ||

Synonym1: | |||

Synonym2: | |||

Synonym3: | |||

Symbol: | |||

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 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 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 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: | 2007-08 | ||

Source: | |||

Replaces: | |||

Internal notes: | |||

CO remarks: | |||

TC/SC remarks: | |||

VT remarks: | |||

Domain1: | |||

Domain2: | |||

Domain3: | |||

Domain4: | |||

Domain5: |

- $V\cdot {U}^{*}={(U\cdot {V}^{*})}^{*}$,
- $(\alpha \text{\hspace{0.17em}}U)\cdot {V}^{*}=\alpha \text{\hspace{0.17em}}(U\cdot {V}^{*})$ 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 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 For two complex vectors or two complex vector quantities ** U** and

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.

102-03-18en.gif |