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 ?
At long last, the assaulter checks whether the 2-Norm of the redesigned tis short of what [[tau].
When matrixes size become larger, like Lena (256 x 256), the advantage of using 2-norm instead of SVD to acquire largest singular value becomes much more explicit.
The 2-norm of the ith column of X, which we denote by [[?
2], equipped with the Euclidean 2-norm [[parallel][x.
x,[infinity]] denote the normalized 2-norms and infinite-norms of the estimation errors ([P.
Clearly, the computations for this formula are much more complex than those for cos [theta](X, Y), so in the 2-norm we are much better off sticking with the latter measure.
8 The condition number estimated using the matrix 2-norm is invariant with respect to orthogonal transformations.
3]VMP) to some existing solvable conic relaxations on the 2-norm soft margin [S.
infinity]] norm expresses the maximum of generalized gain of the system transfer function matrix for a class of input signals characterized by their 2-norm [23].
so that, again from the definition of the matrix 2-norm,
It is a well-known fact about Krylov space solvers that aiming for the smallest 2-norm of the residual, that is, applying GMRes without restarts, is not only excessively memory consuming, but is often also not much faster than using alternative methods that are suboptimal.
Here and [parallel] * [parallel] denote, respectively, the Frobenius and the 2-norm, and [cross product] denotes the Kronecker product.