The direct sum is an
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
between
structures in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
. The direct sum of two abelian groups
and
is another abelian group
consisting of the ordered pairs
where
and
. To add ordered pairs, we define the sum
to be
; in other words addition is defined coordinate-wise. For example, the direct sum
, where
is
real coordinate space
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
, is the
Cartesian plane,
. A similar process can be used to form the direct sum of two
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s or two
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
.
We can also form direct sums with any finite number of summands, for example
, provided
and
are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). This relies on the fact that the direct sum is
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
up to isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. That is,
for any algebraic structures
,
, and
of the same kind. The direct sum is also
commutative up to isomorphism, i.e.
for any algebraic structures
and
of the same kind.
The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
. This is false, however, for some algebraic objects, like nonabelian groups.
In the case where infinitely many objects are combined, the direct sum and direct product are not isomorphic, even for abelian groups, vector spaces, or modules. As an example, consider the direct sum and direct product of (countably) infinitely many copies of the integers. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there is a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. Often, if a + sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be 1. In more technical language, if the summands are
, the direct sum
is defined to be the set of tuples
with
such that
for all but finitely many ''i''. The direct sum
is contained in the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
, but is strictly smaller when the
index set is infinite, because an element of the direct product can have infinitely many nonzero coordinates.
Examples
The ''xy''-plane, a two-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, can be thought of as the direct sum of two one-dimensional vector spaces, namely the ''x'' and ''y'' axes. In this direct sum, the ''x'' and ''y'' axes intersect only at the origin (the zero vector). Addition is defined coordinate-wise, that is
, which is the same as vector addition.
Given two structures
and
, their direct sum is written as
. Given an
indexed family of structures
, indexed with
, the direct sum may be written
. Each ''A
i'' is called a direct summand of ''A''. If the index set is finite, the direct sum is the same as the direct product. In the case of groups, if the group operation is written as
the phrase "direct sum" is used, while if the group operation is written
the phrase "direct product" is used. When the index set is infinite, the direct sum is not the same as the direct product since the direct sum has the extra requirement that all but finitely many coordinates must be zero.
Internal and external direct sums
A distinction is made between internal and external direct sums, though the two are isomorphic. If the summands are defined first, and then the direct sum is defined in terms of the summands, we have an external direct sum. For example, if we define the real numbers
and then define
the direct sum is said to be external.
If, on the other hand, we first define some algebraic structure
and then write
as a direct sum of two substructures
and
, then the direct sum is said to be internal. In this case, each element of
is expressible uniquely as an algebraic combination of an element of
and an element of
. For an example of an internal direct sum, consider
(the integers modulo six), whose elements are
. This is expressible as an internal direct sum
.
Types of direct sum
Direct sum of abelian groups
The direct sum of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s is a prototypical example of a direct sum. Given two such
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
and
their direct sum
is the same as their
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
. That is, the underlying set is the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
and the group operation
is defined component-wise:
This definition generalizes to direct sums of finitely many abelian groups.
For an arbitrary family of groups
indexed by
their
[
is the subgroup of the direct product that consists of the elements that have finite ]support
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
, where by definition, is said to have if is the identity element of for all but finitely many
The direct sum of an infinite family of non-trivial groups is a proper subgroup of the product group
Direct sum of modules
The ''direct sum of modules'' is a construction which combines several modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
into a new module.
The most familiar examples of this construction occur when considering vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, which are modules over a field. The construction may also be extended to Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s and Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s.
Direct sum in categories
An additive category is an abstraction of the properties of the category of modules. In such a category, finite products and coproducts agree and the direct sum is either of them, cf. biproduct.
General case:
In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
the is often, but not always, the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category of modules.
Direct sums versus coproducts in category of groups
However, the direct sum (defined identically to the direct sum of abelian groups) is a coproduct of the groups and in the category of groups. So for this category, a categorical direct sum is often simply called a coproduct to avoid any possible confusion.
Direct sum of group representations
The direct sum of group representations generalizes the direct sum of the underlying modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
, adding a group action to it. Specifically, given a group and two representations and of (or, more generally, two -modules), the direct sum of the representations is with the action of given component-wise, that is,
Another equivalent way of defining the direct sum is as follows:
Given two representations and the vector space of the direct sum is and the homomorphism is given by where is the natural map obtained by coordinate-wise action as above.
Furthermore, if are finite dimensional, then, given a basis of , and are matrix-valued. In this case, is given as
Moreover, if we treat and as modules over the group ring , where is the field, then the direct sum of the representations and is equal to their direct sum as modules.
Direct sum of rings
Some authors will speak of the direct sum of two rings when they mean the direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
, but this should be avoided since does not receive natural ring homomorphisms from and : in particular, the map sending to is not a ring homomorphism since it fails to send 1 to (assuming that in ). Thus is not a coproduct in the category of rings, and should not be written as a direct sum. (The coproduct in the category of commutative rings is the tensor product of rings
In mathematics, the tensor product of two algebras over a commutative ring ''R'' is also an ''R''-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the produ ...
.[, section I.11] In the category of rings, the coproduct is given by a construction similar to the free product of groups.)
Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If is an infinite collection of nontrivial rings, then the direct sum of the underlying additive groups can be equipped with termwise multiplication, but this produces a rng, that is, a ring without a multiplicative identity.
Direct sum of matrices
For any arbitrary matrices and , the direct sum is defined as the block diagonal matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original ma ...
of and if both are square matrices (and to an analogous block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...
, if not).
Direct sum of topological vector spaces
A topological vector space (TVS) such as a Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, is said to be a of two vector subspaces and if the addition map
is an isomorphism of topological vector spaces (meaning that this linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
is a bijective homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
), in which case and are said to be in
This is true if and only if when considered as additive topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s (so scalar multiplication is ignored), is the topological direct sum of the topological subgroups and
If this is the case and if is Hausdorff then and are necessarily closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
subspaces of
If is a vector subspace of a real or complex vector space then there always exists another vector subspace of called an such that is the of and (which happens if and only if the addition map is a vector space isomorphism).
In contrast to algebraic direct sums, the existence of such a complement is no longer guaranteed for topological direct sums.
A vector subspace of is said to be a () if there exists some vector subspace of such that is the topological direct sum of and
A vector subspace is called if it is not a complemented subspace.
For example, every vector subspace of a Hausdorff TVS that is not a closed subset is necessarily uncomplemented.
Every closed vector subspace of a Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is complemented.
But every Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
that is not a Hilbert space necessarily possess some uncomplemented closed vector subspace.
Homomorphisms
The direct sum comes equipped with a '' projection'' homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
for each ''j'' in ''I'' and a ''coprojection'' for each ''j'' in ''I''. Given another algebraic structure (with the same additional structure) and homomorphisms for every ''j'' in ''I'', there is a unique homomorphism , called the sum of the ''g''''j'', such that for all ''j''. Thus the direct sum is the coproduct in the appropriate category.
See also
* Direct sum of groups
*Direct sum of permutations In combinatorics, the skew sum and direct sum of permutations are two operations to combine shorter permutations into longer ones. Given a permutation ''π'' of length ''m'' and the permutation ''σ'' of length ''n'', the skew sum of ''π'' and '' ...
* Direct sum of topological groups
* Restricted product
* Whitney sum
Notes
References
*{{Lang Algebra, edition=3r
Abstract algebra