Euclidean Norm

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A Euclidean Norm is a norm distance between two points in the euclidean space.

  • Context:
    • It can be denoted as [math]L^2[/math] norm.
    • It can be found out for a vector [math]p=\begin{bmatrix} p_1\\ p_2\\ \vdots \\p_n\end{bmatrix}[/math] as [math]\|p\|=\sqrt{{p_1}^2+{p_2}^2+\dots++{p_n}^2}=\sqrt{p.p}[/math]
    • It can be found out for a vector [math]\overrightarrow{pq}[/math] in n-dimensional space where the points [math]p=(p_1,p_2,\dots,p_n)[/math] and [math]q=(q_1,q_2,\dots,q_n)[/math] as [math]\|\boldsymbol{q-p}\|=\sqrt{{q_1-p_1}^2+{q_2-p_2}^2+\dots++{q_n-p_n}^2}=\sqrt{(p-q).(p-q)}[/math]
  • See: n-Dimensional Euclidean Space, Vector Length, Euclidean Distance, Radial Basis Function.



    • On an n-dimensional Euclidean space Rn, the intuitive notion of length of the vector x = (x1, x2, ..., xn) is captured by the formula

       :[math]\|\boldsymbol{x}\| := \sqrt{x_1^2 + \cdots + x_n^2}.[/math]

      This gives the ordinary distance from the origin to the point x, a consequence of the Pythagorean theorem.

      The Euclidean norm is by far the most commonly used norm on Rn, but there are other norms on this vector space as will be shown below. However all these norms are equivalent in the sense that they all define the same topology.

      On an n-dimensional complex space Cn the most common norm is :[math]\|\boldsymbol{z}\| := \sqrt{|z_1|^2 + \cdots + |z_n|^2}= \sqrt{z_1 \bar z_1 + \cdots + z_n \bar z_n}.[/math] In both cases we can also express the norm as the square root of the inner product of the vector and itself: :[math]\|\boldsymbol{x}\| := \sqrt{\boldsymbol{x}^* ~ \boldsymbol{x}},[/math] where x is represented as a column vector ([x1; x2; ...; xn]), and x* denotes its conjugate transpose.

      This formula is valid for any inner product space, including Euclidean and complex spaces. For Euclidean spaces, the inner product is equivalent to the dot product. Hence, in this specific case the formula can be also written with the following notation: :[math]\|\boldsymbol{x}\| := \sqrt{\boldsymbol{x} \cdot \boldsymbol{x}}.[/math] The Euclidean norm is also called the 'Euclidean length, L2 distance, 2 distance, L2 norm, or 2 norm ; see Lp space.

      The set of vectors in Rn+1 whose Euclidean norm is a given positive constant forms an n-sphere.