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Frobenius Norm


The Frobenius norm, sometimes also called the Euclidean norm (a term unfortunately also used for the vector L^2-norm), is matrix norm of an m×n matrix A defined as the square root of the sum of the absolute squares of its elements,

 ||A||_F=sqrt(sum_(i=1)^msum_(j=1)^n|a_(ij)|^2)

(Golub and van Loan 1996, p. 55).

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.,

 ||A||_F=sqrt(Tr(AA^(H))).

The Frobenius norm of a matrix m is implemented as Norm[m, "Frobenius"] and of a vector v as Norm[v, "Frobenius"].


See also

Hilbert-Schmidt Norm, Matrix Norm

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References

Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, 1996.Higham, N. J. "Matrix Norms." §6.2 in Accuracy and Stability of Numerical Algorithms. Philadelphia: Soc. Industrial and Appl. Math., 1996.Horn, R. A. and Johnson, C. R. "Norms for Vectors and Matrices." Ch. 5 in Matrix Analysis. Cambridge, England: Cambridge University Press, 1990.

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Frobenius Norm

Cite this as:

Weisstein, Eric W. "Frobenius Norm." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FrobeniusNorm.html

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