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


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


where A* denotes the conjugate transposeMathworldPlanetmath of A.

Some :

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

  • Let A be a square matrixMathworldPlanetmath and let U be a unitary matrixMathworldPlanetmath of same size as A. Then ||A||F=||UAU||F where U is the conjugate transpose of U.

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

Title Frobenius matrix norm
Canonical name FrobeniusMatrixNorm
Date of creation 2013-03-22 11:43:25
Last modified on 2013-03-22 11:43:25
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 25
Author mathcam (2727)
Entry type Definition
Classification msc 65F35
Classification msc 15A60
Classification msc 18-00
Synonym Euclidean matrix norm
Synonym matrix F-norm
Synonym Hilbert-Schmidt norm
Related topic MatrixNorm
Related topic MatrixPnorm
Related topic VectorNorm
Related topic VectorPnorm
Related topic ShursInequality
Related topic trace
Related topic transposeMathworldPlanetmath
Related topic Transpose
Related topic MatrixLogarithm
Related topic FrobeniusProduct