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space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces...

the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or,...

+(p_{n}-q_{n})^{2}}}.} The Euclidean distance may also be expressed more compactly in terms of the Euclidean norm of the Euclidean vector difference: d (...

specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows...

{\displaystyle p=2} (the Euclidean norm or ℓ 2 {\displaystyle \ell _{2}} -norm for vectors), the induced matrix norm is the spectral norm. The two values do...

the measure of units between a number and zero. In vector spaces, the Euclidean norm is a measure of magnitude used to define a distance between two points...

norm. An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm of...

known as the norm squared, a ⋅ a = ‖ a ‖ 2 {\textstyle \mathbf {a} \cdot \mathbf {a} =\|\mathbf {a} \|^{2}} , after the Euclidean norm; it is a vector...

(see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The...

corresponds to a norm function, such as that used to order the Gaussian integers above, then the domain is known as norm-Euclidean. The norm-Euclidean rings of...

basis for the vector space V and a norm N. The norm usually considered is the Euclidean norm L2. However, other norms (such as Lp) are also considered and...

in infinite-dimensional vector spaces.) The dual of the Euclidean norm is the Euclidean norm, since sup { z ⊺ x : ‖ x ‖ 2 ≤ 1 } = ‖ z ‖ 2 . {\displaystyle...

Euclidean space R n {\displaystyle \mathbb {R} ^{n}} by the Euclidean metric. The Euclidean norm on R n {\displaystyle \mathbb {R} ^{n}} is the non-negative...

matrices of real numbers; these induced norms form a subset of matrix norms. If we specifically choose the Euclidean norm on both R n {\displaystyle \mathbb...

real vector space R n {\displaystyle \mathbb {R} ^{n}} is given by the Euclidean norm: ‖ x ‖ 2 = ( x 1 2 + x 2 2 + ⋯ + x n 2 ) 1 / 2 . {\displaystyle...

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric...

distinct notation is the use of vertical bars for either the Euclidean norm or sup norm of a vector in R n {\displaystyle \mathbb {R} ^{n}} , although...

This is always a non-negative real number, and it is the same as the Euclidean norm on H {\displaystyle \mathbb {H} } considered as the vector space R 4...

coordinates of the points of a Euclidean space of dimension n, En (Euclidean line, E; Euclidean plane, E2; Euclidean three-dimensional space, E3) form...

{\displaystyle v_{1}\in \mathbb {C} ^{n}} be an arbitrary vector with Euclidean norm 1 {\displaystyle 1} . Abbreviated initial iteration step: Let w 1 ′...