Euclidean Space

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euclidean space

[yü′klid·ē·ən ′spās]
A space consisting of all ordered sets (x1, …, xn ) of n numbers with the distance between (x1, …, xn ) and (y1, …, yn ) being given by the number n is called the dimension of the space.

Euclidean Space


in mathematics, a space whose properties are described by the axioms of Euclidean geometry. In a more general sense, a Euclidean space is an n-dimensional vector space, into which several special Cartesian coordinates can be introduced so that its metric is defined in the following manner: If point M has the coordinates (x1x2, …, xn and point M* has the coordinates (x1*, x2*, …, xn*), then the distance between these points is

References in periodicals archive ?
The 2-norm of the ith column of X, which we denote by [[?
In the previous sections, we derived the bound on the orthogonality error in terms of the 2-norm, because we wanted to give a bound on the 2-norm condition number of [?
2], equipped with the Euclidean 2-norm [[parallel][x.
Then F is called a probabilistic 2-norm and (X, F, [tau]) a probabilistic 2-normed space if the following conditions are satisfied:
x,[infinity]] denote the normalized 2-norms and infinite-norms of the estimation errors ([P.
so that, again from the definition of the matrix 2-norm,
where [kappa](S) is the condition number of S with respect to the 2-norm.
i] are not orthogonal any more, an increase (decrease) in the coefficients does not necessarily imply an increase (decrease) of the 2-norm.
iv) the standard 2-norm stopping criterion [parallel][r.
In particular, we derive six separate algebraic bounds on the 2-norm of a real matrix; to the best of our knowledge, these bounds are new.
k,i] and post-multiplication by an orthogonal matrix does not affect the 2-norms of rows of [A.