请输入您要查询的字词:

 

单词 FrobeniusMatrixNorm
释义

Frobenius matrix norm


Let R be a ring with a valuation || and let M(R) denote the set of matrices over R. The Frobenius normMathworldPlanetmath function or Euclidean matrix norm is the norm function ||||F:M(R) given by

||A||F=i=1mj=1n|aij|2,

where m and n respectively denote the number of rows and columns of A. Note A need not be square for this definition. A more concise (though ) definition, in the case that R= or , is

||A||F=trace(A*A),

where A* denotes the conjugate transposeMathworldPlanetmath of A.

Some :

  • Denote the columns of A by Ai. A nice property of the norm is that

    ||A||F2=||A1||22+||A2||22++||An||22.
  • Let A be a square matrixMathworldPlanetmath and let U be a unitary matrixMathworldPlanetmathof same size as A. Then ||A||F=||UAU||Fwhere U is the conjugate transpose of U.

  • If AB is defined, then ||AB||F||A||F||B||F.

TitleFrobenius matrix norm
Canonical nameFrobeniusMatrixNorm
Date of creation2013-03-22 11:43:25
Last modified on2013-03-22 11:43:25
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id25
Authormathcam (2727)
Entry typeDefinition
Classificationmsc 65F35
Classificationmsc 15A60
Classificationmsc 18-00
SynonymEuclidean matrix norm
Synonymmatrix F-norm
SynonymHilbert-Schmidt norm
Related topicMatrixNorm
Related topicMatrixPnorm
Related topicVectorNorm
Related topicVectorPnorm
Related topicShursInequality
Related topictrace
Related topictransposeMathworldPlanetmath
Related topicTranspose
Related topicMatrixLogarithm
Related topicFrobeniusProduct
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 22:47:48