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IEVref:102-03-23ID:
Language:enStatus: Standard
Term: magnitude, <of a vector>
Synonym1: norm, <of a vector>
Synonym2:
Synonym3:
Symbol:
Definition: for any vector U, non-negative scalar, usually denoted by |U| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvaa Gaay5bSlaawIa7aaaa@3CCD@ , equal to the non-negative square root of the scalar product or, in the case of a complex vector, of the Hermitian product of the vector by itself

Note 1 to entry: The magnitude of a vector U has the following properties:

  • U=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGH9aqpie qajugqbiaa=bdaaaa@3C1B@ if and only if |U|=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvaa Gaay5bSlaawIa7aiabg2da9iaaicdaaaa@3E8D@ ,
  • | αU|=|α||U| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaeqySde MaaCyvaaGaay5bSlaawIa7aiabg2da9maaemaabaGaeqySdegacaGL hWUaayjcSdGaeyyXIC9aaqWaaeaacaWHvbaacaGLhWUaayjcSdaaaa@4A7D@ where α is a scalar,
  • | U+V||U|+|V| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvai abgUcaRiaahAfaaiaawEa7caGLiWoacqGHKjYOdaabdaqaaiaahwfa aiaawEa7caGLiWoacqGHRaWkdaabdaqaaiaahAfaaiaawEa7caGLiW oaaaa@4926@ where V is any other vector.

Note 2 to entry: For a vector U in the three-dimensional Euclidean or Hermitian space with orthonormal base, the magnitude is given by |U|= | U 1 | 2 + | U 2 | 2 + | U 3 | 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaemaabaGaaCyvaa Gaay5bSlaawIa7aiabg2da9maakaaabaWaaqWaaeaacaWGvbWaaSba aSqaaiaaigdaaeqaaaGccaGLhWUaayjcSdWaaWbaaSqabeaacaaIYa aaaOGaey4kaSYaaqWaaeaacaWGvbWaaSbaaSqaaiaaikdaaeqaaaGc caGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaqWaae aacaWGvbWaaSbaaSqaaiaaiodaaeqaaaGccaGLhWUaayjcSdWaaWba aSqabeaacaaIYaaaaaqabaaaaa@5140@ .

Note 3 to entry: The terms "Euclidean norm" and "Hermitian norm" may be used for the real or the complex case, respectively.

Note 4 to entry: The magnitude of a vector U is represented by |U| MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaa4WaaqWaaOqaaKqzaf GaaCyvaaGccaGLhWUaayjcSdaaaa@3D9E@ or by U; U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaa4WaauWaaOqaaKqzaf GaaCyvaaGccaGLjWUaayPcSdaaaa@3DA3@ is also used.


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
CO remarks:
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VT remarks:
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