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Vector Norm -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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Given an n-dimensional vector

x=[x_1; x_2; |; x_n],

(1)

a general vector norm |x|, sometimes written with a double bar as ||x||, is a nonnegative norm defined such that

1. |x|>0 when x!=0 and |x|=0 iff x=0.

2. |kx|=|k||x| for any scalar k.

3. |x+y|<=|x|+|y|.

In this work, a single bar is used to denote a vector norm, absolute value, or complex modulus, while a double bar is reserved for denoting a matrix norm.

The vector norm |x|_p for p=1, 2, ... is defined as

|x|_p=(sum_(i)|x_i|^p)^(1/p).

(2)

The p-norm of vector v is implemented as Norm[v, p], with the 2-norm being returned by Norm[v].

The special case |x|_infty is defined as

|x|_infty=max_(i)|x_i|.

(3)

The most commonly encountered vector norm (often simply called "the norm" of a vector, or sometimes the magnitude of a vector) is the L2-norm, given by

|x|_2=|x|=sqrt(x_1^2+x_2^2+...+x_n^2).

(4)

This and other types of vector norms are summarized in the following table, together with the value of the norm for the example vector v=(1,2,3).

See also

Compatible, Distance, Euclidean Metric, L1-Norm, L2-Norm, L-infty-Norm, Matrix Norm, Natural Norm, Norm, Vector Magnitude

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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. 1114, 2000.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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Vector Norm

Cite this as:

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

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