The norm of a mathematical object is a quantity that in some (possibly abstract) sense describes the length, size, or extent of the object. Norms exist for complex numbers (the complex modulus, sometimes also called the complex norm or simply "the norm"), Gaussian integers (the same as the complex modulus, but sometimes unfortunately instead defined to be the absolute square), quaternions (quaternion norm), vectors (vector norms), and matrices (matrix norms). A generalization of the absolute value known as the p-adic norm is also defined.

On a real vector space or complex vector space, a vector norm assigns a nonnegative real number to each vector. It vanishes iff the vector is the zero vector, satisfies for every scalar , and obeys the triangle inequality (Axler 2026, pp. 186 and 190).

Norms are variously denoted , , , or . In this work, single bars are used to denote the complex modulus, quaternion norm, p-adic norms, and vector norms, while the double bar is reserved for matrix norms.

The term "norm" is often used without additional qualification to refer to a particular type of norm (such as a matrix norm or vector norm). Most commonly, the unqualified term "norm" refers to the flavor of vector norm technically known as the L²-norm. This norm is variously denoted , , or , and gives the length of an n-vector . It can be computed as

The complex modulus, 2-norm of a vector, or 2-norm of a matrix is returned by Norm[expr]. The -norm of a vector is returned by Norm[v, p] for a real number , with Infinity also supported. For a matrix, the documented specifications are 1, 2, Infinity, and "Frobenius", giving the maximum absolute column sum norm, spectral norm, maximum absolute row sum norm, and Frobenius norm, respectively.

The norm (length) of a vector should not be confused with a normal vector (a vector perpendicular to a surface).

See also

Bombieri Norm, Compatible, Complex Modulus, Complex Number, Field Norm, Four-Vector Norm, Frobenius Norm, Hilbert-Schmidt Norm, L¹-Norm, L²-Norm, L^∞-Norm, Matrix Norm, Maximum Absolute Column Sum Norm, Maximum Absolute Row Sum Norm, Natural Norm, Normal Vector, Normalized Vector, Normed Space, Parallelogram Law, Polynomial Norm, Quasi-Norm, Quaternion Norm, Spectral Norm, Vector Norm Explore this topic in the MathWorld classroom

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References

Axler, S. Linear Algebra Done Right, 4th ed. Author's version, Aug. 16, 2026. https://linear.axler.net/LADR4e.pdf.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1114-1125, 2000.

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Norm

Cite this as:

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

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