componentwise

We could, for example, try to define a relation between sequences in a componentwise fashion:

If we denote the -th component of any vector as , then componentwise addition is .

This ordering is called the product order, or alternatively the coordinatewise order, or even the componentwise order.

Complex coordinate space is a vector space over the complex numbers, with componentwise addition and scalar multiplication.

In the category of modules over some ring R, the product is the cartesian product with addition defined componentwise and distributive multiplication.

This can be used to generalize for vector valued functions (f : U R) by carefully using a componentwise argument.

Componentwise operations are usually defined on vectors, where vectors are elements of the set for some natural number and some field .

The set of such Cauchy sequences forms a group (for the componentwise product), and the set of null sequences (s.th. )

The same construction also works for an arbitrary family of rings: if are rings indexed by a set I, then is a ring with componentwise addition and multiplication.

The space H of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication:

Therefore any vector corresponds to the function such that , and any componentwise operation on vectors is the pointwise operation on functions corresponding to those vectors.

If p, q, r are partitions then if and only if The componentwise minimum of two nondecreasing concave integer sequences is also nondecreasing and concave.

DESCRIPTION Componentwise scaling of a vector to specified output ranges can be achieved with AutoScaler.

Therefore, for any two partitions of n, p and q, their meet is the partition of n whose associated (n + 1)-tuple has components The natural idea to use a similar formula for the join fails, because the componentwise maximum of two concave sequences need not be concave.

If, for example in a 2-dimensional Euclidean space, the new basis vectors are rotated anti-clockwise with respect to the old basis vectors, then it will appear in terms of the new system that the componentwise representation of the vector was rotated in the opposite direction, i.e. clockwise (see figure).

However, the existence of a componentwise derivative does not guarantee the existence of a derivative, as componentwise convergence in a Hilbert space does not guarantee convergence with respect to the actual topology of the Hilbert space.