Frobenius Norm

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



References

2015

[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, σi 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

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]