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Natural Norm

A natural norm of an n×n matrix A, with n>=1, is the matrix norm induced, or subordinate, to a chosen vector norm on C^n or R^n:

||A||=max_(||z||=1)||Az||.

The same vector norm is used for z and Az. For any natural norm,

||I||=1,

where I is the identity matrix. A Frobenius norm is not a natural norm when n>1, since its value on the identity matrix is sqrt(n) rather than 1 (Higham 2021). The natural matrix norms induced by the L1-norm, L2-norm, and L∞-norm are called the maximum absolute column sum norm, spectral norm, and maximum absolute row sum norm, respectively.

See also

L1-Norm, L2-Norm, Matrix Norm, Maximum Absolute Column Sum Norm, Spectral Norm, Vector Norm

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1115, 2000.Higham, N. J. "What Is a Matrix Norm?" Oct. 12, 2021. https://nhigham.com/2021/10/12/what-is-a-matrix-norm/.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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Natural Norm

Cite this as:

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

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