Natural Norm

Let |z| be a vector norm of a vector z such that


Then ||A|| is a matrix norm which is said to be the natural norm induced (or subordinate) to the vector norm |z|. For any natural norm,


where I is the identity matrix. The natural matrix norms induced by the L1-norm, L2-norm, and L-infty-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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Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1115, 2000.Horn, R. A. and Johnson, C. R. "Norms for Vectors and Matrices." Ch. 5 in Matrix Analysis. Cambridge, England: Cambridge University Press, 1990.

Referenced on Wolfram|Alpha

Natural Norm

Cite this as:

Weisstein, Eric W. "Natural Norm." From MathWorld--A Wolfram Web Resource.

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