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Frobenius matrix norm (Definition)

Let $ R$ be a ring with a valuation $ \vert\cdot\vert$ and let $ M(R)$ denote the set of matrices over $ R$. The Frobenius norm function or Euclidean matrix norm is the norm function $ \vert\vert\,\cdot\,\vert\vert _F:M(R)\rightarrow \mathbb{R}$ given by

$\displaystyle \vert\vert\,A\,\vert\vert _F = \sqrt{\sum_{i=1}^m\sum_{j=1}^n\vert a_{ij}\vert^2},$    

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 equivalent) definition, in the case that $ R=\mathbb{R}$ or $ \mathbb{C}$, is
$\displaystyle \vert\vert\,A\,\vert\vert _F = \sqrt{\textrm{trace}(A^*A)},$    

where $ A^*$ denotes the conjugate transpose of $ A$.

Some properties:

  • Denote the columns of $ A$ by $ A_i$. A nice property of the norm is that
    $\displaystyle \vert\vert A\vert\vert _F^2=\vert\vert A_1\vert\vert _2^2+\vert\vert A_2\vert\vert _2^2+\cdots+\vert\vert A_n\vert\vert _2^2.$    

  • Let $ A$ be a square matrix and let $ U$ be a unitary matrix of same size as $ A$. Then $ \vert\vert\,A\,\vert\vert _F = \vert\vert\,U^\ast A U\,\vert\vert _F$ where $ U^\ast$ is the conjugate transpose of $ U$.
  • If $ AB$ is defined, then $ \vert\vert\,A B\,\vert\vert _F \le \vert\vert\,A\,\vert\vert _F\ \vert\vert\,B\,\vert\vert _F$.



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See Also: matrix norm, matrix p-norm, vector norm, vector p-norm, Schur's inequality, trace, transpose, transpose, matrix logarithm, Frobenius product

Other names:  Euclidean matrix norm, matrix F-norm, Hilbert-Schmidt norm

Attachments:
Schur's inequality (Theorem) by matte
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Cross-references: size, unitary matrix, square matrix, property, conjugate transpose, square, columns, rows, number, norm, function, Frobenius norm, matrices, valuation, ring
There are 7 references to this entry.

This is version 15 of Frobenius matrix norm, born on 2001-10-06, modified 2007-06-24.
Object id is 109, canonical name is FrobeniusMatrixNorm.
Accessed 24927 times total.

Classification:
AMS MSC: 15A60 (Linear and multilinear algebra; matrix theory :: Norms of matrices, numerical range, applications of functional analysis to matrix theory)
 65F35 (Numerical analysis :: Numerical linear algebra :: Matrix norms, conditioning, scaling)
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