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Norm (mathematics)

From Norm (mathematics) - Wikipedia:

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin.

Typically, for \(\boldsymbol{x} = (x_1, x_2, \cdots, x_n) \in \mathbb{R}^n\), the norm of \(x\) is defined as

\[ || \boldsymbol{x} || = \left( x_1^2 + \cdots + x_n^2 \right)^{\frac{1}{2}} \]

This definition of norm is called the Euclidean norm or the \(L^2\) norm

Similarly, for \(\boldsymbol{z} = (z_1, z_2, \cdots, z_n) \in \mathbb{C}^n\), the norm of \(\boldsymbol{z}\) is defined as

\[ || \boldsymbol{z} || = \left( |z_1|^2 + \cdots + |z_n|^2 \right)^{\frac{1}{2}} \]

where \(|z_i|\) is the absolute value of \(z_i\).

There are also other definitions of norm. For example, the \(L^p\) norm for \(\boldsymbol{z}\) is defined as:

\[ || \boldsymbol{z} ||_p = \left( |z_1|^p + \cdots + |z_n|^p \right)^{\frac{1}{p}} \]

Norm of complex number

In some fields or some applications, the norm of a complex number \(z = a + ib\) is defined as \((a^2 + b^2)\) instead of \(\sqrt{(a^2 + b^2)}\) . See Gauss' definition of norm for complex number and std::norm in C++ standard library.

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