Frobenius Norm

A Frobenius Norm is a matrix norm which is equal to square root of the matrix trace.

• AKA: Hilbert–Schmidt Norm.
• Context:
  • It is defined as [math]\displaystyle{ \|A\|_F=\sqrt{\sum_{i=1}^m\sum_{j=1}^n |a_{ij}|^2} }[/math].
  • It can be defined simply as the Euclidean Norm.
  • It can be also defined as Schatten Norm with p=2.
  • It can also be a vector norm.
  • …
• Example(s):
  • Euclidean Norm.
  • Schatten Norm.
• Counter-Example(s):
  • Trace Norm.
  • Max Norm.
• See: Matrix Norm, Trace Norm, P-Norm.

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References

2015

• (Wikipedia, 2015) ⇒ http://wikipedia.org/wiki/Matrix_norm#Frobenius_norm
  • QUOTE: When p = q = 2 for the [math]\displaystyle{ L_{p,q} }[/math] norm, it is called the Frobenius norm or the Hilbert–Schmidt norm, though the latter term is often reserved for operators on Hilbert space. This norm can be defined in various ways:

[math]\displaystyle{ \|A\|_F=\sqrt{\sum_{i=1}^m\sum_{j=1}^n |a_{ij}|^2}=\sqrt{\operatorname{trace}(A^{{}^*}A)}=\sqrt{\sum_{i=1}^{\min\{m,\,n\}} \sigma_i^2} }[/math]

where A^* denotes the conjugate transpose of A, σ_i are the singular values of A, and the trace function is used. The Frobenius norm is similar to the Euclidean norm on Kⁿ and comes from the Frobenius inner product on the space of all matrices.

The Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. This norm is often easier to compute than induced norms and has the useful property of being invariant under rotations, that is, [math]\displaystyle{ \|A\|_F^{2}=\|AR\|_F^{2}=\|RA\|_F^{2} }[/math] for any rotation matrix [math]\displaystyle{ R }[/math]. This property follows from the trace definition restricted to real matrices,

[math]\displaystyle{ \|AR\|_F^{2} = \operatorname{trace} \left( R^{\rm T} A^{\rm T}A R \right) = \operatorname{trace}\left( RR^{\rm T} A^{\rm T}A \right) = \operatorname{trace}\left(A^{\rm T}A \right) = \|A\|_F^{2} }[/math]

and

[math]\displaystyle{ \|RA\|_F^{2} = \operatorname{trace} \left( A^{\rm T} R^{\rm T} RA \right) = \operatorname{trace}\left(A^{\rm T}A \right) = \|A\|_F^2 }[/math]

where we have used the orthogonal nature of [math]\displaystyle{ R }[/math], that is, [math]\displaystyle{ R^{\rm T} R = R R^{\rm T} = \mathbf{I} }[/math], and the cyclic nature of the trace, [math]\displaystyle{ \operatorname{trace}(XYZ) = \operatorname{trace}(ZXY) }[/math]. More generally the norm is invariant under a unitary transformation for complex matrices.

2012

• (Li et al., 2012) ⇒ Liangda Li, Guy Lebanon, and Haesun Park. (2012). “Fast Bregman Divergence NMF Using Taylor Expansion and Coordinate Descent.” In: Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD-2012). ISBN:978-1-4503-1462-6 doi:10.1145/2339530.2339582

1999

• (Wolfram Mathworld , 1999) ⇒ http://mathworld.wolfram.com/FrobeniusNorm.html
  • QUOTE: The Frobenius norm, sometimes also called the Euclidean norm (which may cause confusion with the vector L^2-norm which also sometimes known as the Euclidean norm), is matrix norm of an [math]\displaystyle{ m \times n }[/math] matrix A defined as the square root of the sum of the absolute squares of its elements,

[math]\displaystyle{ ||A||_F=\sqrt{\sum_{i=1}^m\sum_{j=1}^n|a_{ij}|^2} }[/math]

The Frobenius norm can also be considered as a vector norm.

It is also equal to the square root of the matrix trace of AA^H, where A^H is the conjugate transpose, i.e.,

[math]\displaystyle{ ||A||_F=\sqrt{Tr(AA^H)} }[/math]

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