Direct sum

This article deals with direct sums of vector spaces, for direct sums of permutations see Sum of Permutations .

In mathematics, the term “ direct sum ” denotes the outer direct sum and the inner direct sum .

In both cases, the direct sum is written with the link sign (circled plus sign , Unicode : U + 2295 circled plus sign , or as a multi-digit operator analogous to the summation sign : U + 2A01 n-ary circled plus operator ). ${\displaystyle \oplus }$

External direct sum

In mathematics, the external direct sum is the standard representative of the coproduct of Abelian groups or modules (and thus also vector spaces ) defined in category theory (only except for isomorphism ). It is given by the subgroup or sub-module of the direct product , which consists of the tuples with at most a finite number of entries different from the (respective) zero element. In the case of a finite number of factors, this structure obviously corresponds to the direct product. (In the following, for the sake of simplicity, we will only deal with the case of vector spaces, but this is analogous for the direct sum of Abelian groups and the direct sum of modules.)

Another way of describing the coproduct is the inner direct sum explained below , which is isomorphic to the outer direct sum.

definition

Be a body and a family of vector spaces. Then is called ${\displaystyle K}$ ${\displaystyle \left(V_{i}\right)_{i\in I}}$ ${\displaystyle K}$

${\displaystyle \bigoplus _{i\in I}V_{i}:={\Big \{}\left(v_{i}\right)_{i\in I}\in \prod _{i\in I}V_{i}\;{\Big |}\;v_{i}=0}$for almost everyone ${\displaystyle i\in I\ {\Big \}}}$

is the outer direct sum of the family , where is the direct product of vector spaces. ${\displaystyle (V_{i})_{i\in I}}$ ${\displaystyle \textstyle \prod _{i\in I}V_{i}}$

In the finite case, for example, this results

${\displaystyle V_{1}\oplus V_{2}=\left\{\left(v_{1},v_{2}\right)\mid v_{1}\in V_{1},v_{2}\in V_{2}\right\}=V_{1}\times V_{2}}$

The distinction between direct sum and direct product is therefore only necessary if the index amount is infinite.

In addition, for such a direct sum of finitely many vector spaces, the dimension of the sum is equal to the sum of the dimensions of its summands.

Inner direct sum

In the case of a family of subspaces of the vector space , the inner direct sum is called the (they are then also called the direct decomposition of ), if each (apart from the sequence) is uniquely the sum of finitely many elements of the subspaces, with at most one element from each subspace and never the zero element is selected, can be displayed, d. H.: ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle U_{i}}$ ${\displaystyle U_{i}}$ ${\displaystyle V}$ ${\displaystyle v\in V}$

For every vector there is exactly one family of vectors with for all and only for finitely many of , so that is. ${\displaystyle v\in V}$ ${\displaystyle (u_{i})_{i\in I}}$ ${\displaystyle u_{i}\in U_{i}}$ ${\displaystyle i\in I}$ ${\displaystyle u_{i}\neq 0}$ ${\displaystyle u_{i}}$ ${\displaystyle v=\sum _{i\in I}u_{i}}$

Like the outer sum, the inner sum is symbolized as follows:

${\displaystyle V=\bigoplus _{i\in I}U_{i}}$

or in the finite case

${\displaystyle V=U_{1}\oplus \dotsb \oplus U_{n}}$.

A sum of a family of subspaces is direct if and only if it holds for all : ${\displaystyle V=\sum _{i\in I}U_{i}}$ ${\displaystyle j\in I}$

${\displaystyle U_{j}\cap \sum _{i\in I\setminus \{j\}}U_{i}=\{0\}}$,

that is, if for each the intersection with the sum of the remaining sub-vector spaces contains only the zero vector. ${\displaystyle U_{j}}$

In special cases one calls and complementary to each other . The following applies ${\displaystyle U_{1}\oplus U_{2}=V}$ ${\displaystyle U_{1}}$ ${\displaystyle U_{2}}$

${\displaystyle U_{1}\oplus U_{2}=V\Leftrightarrow (U_{1}+U_{2}=V)\land (U_{1}\cap U_{2}=\{0\})}$.

A subspace of a vector space is called a direct summand of if there is a subspace that is too complementary, i.e. for which applies. ${\displaystyle U_{1}\subset V}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle U_{1}}$ ${\displaystyle U_{2}}$ ${\displaystyle U_{1}\oplus U_{2}=V}$

context

Note: The outer sum of subspaces can always be formed, but the inner sum of subspaces is usually not direct.

The relationship between the inner and outer total can be established as follows.

For each, consider the embedding in the outer direct sum, so: ${\displaystyle j\in I}$ ${\displaystyle f_{j}\colon V_{j}\longrightarrow \oplus V_{i}}$

${\displaystyle f_{j}(x)=(v_{i})_{i\in I},v_{i}=x}$for and for ${\displaystyle i=j}$ ${\displaystyle v_{i}=0}$ ${\displaystyle i\neq j}$

The inner direct sum of the images in these images then forms the outer direct sum.

Direct sum of representations

Be depictions of or the direct sum of the representations is defined as: being for all and in this way is again a linear representation. If representations of the same group are defined, for the sake of simplicity, the direct sum of the representations is also defined as representation of, i.e. in which one understands as the diagonal subgroup of . ${\displaystyle \textstyle (\rho _{1},V_{\rho _{1}}),(\rho _{2},V_{\rho _{2}})}$ ${\displaystyle \textstyle G_{1}}$ ${\displaystyle \textstyle G_{2}.}$ ${\displaystyle \textstyle \rho _{1}\oplus \rho _{2}:G_{1}\times G_{2}\to {\text{GL}}(V_{\rho _{1}}\oplus V_{\rho _{2}}),}$ ${\displaystyle \textstyle \rho _{1}\oplus \rho _{2}(s_{1},s_{2})(v_{1},v_{2}):=\rho _{1}(s_{1})v_{1}\oplus \rho _{2}(s_{2})v_{2}}$ ${\displaystyle (s_{1},s_{2})\in G_{1}\times G_{2}}$ ${\displaystyle v_{1}\in V_{\rho _{1}},v_{2}\in V_{\rho _{2}}.}$
${\displaystyle \textstyle \rho _{1}\oplus \rho _{2}}$
${\displaystyle \textstyle \rho _{1},\rho _{2}}$ ${\displaystyle \textstyle G,}$ ${\displaystyle \textstyle G,}$ ${\displaystyle \textstyle \rho _{1}\oplus \rho _{2}\colon G\to {\text{GL}}(V_{1}\oplus V_{2}),}$ ${\displaystyle \textstyle G}$ ${\displaystyle \textstyle G\times G}$

example

Let be the linear representation given by ${\displaystyle \textstyle \rho _{1}\colon \mathbb {Z} /2\mathbb {Z} \to {\text{GL}}_{2}(\mathbb {C} )}$

${\displaystyle \rho _{1}({\overline {1}})=\left({\begin{array}{cc}0&-i\\i&0\end{array}}\right).}$

And be the linear representation that is given by ${\displaystyle \textstyle \rho _{2}:\mathbb {Z} /3\mathbb {Z} \to {\text{GL}}_{3}(\mathbb {C} )}$

${\displaystyle \rho _{2}({\overline {1}})=\left({\begin{array}{ccc}1&0&e^{\frac {2\pi i}{3}}\\0&e^{\frac {2\pi i}{3}}&0\\0&0&e^{\frac {4\pi i}{3}}\end{array}}\right).}$

Then is a linear representation of in which the for by definition looks like this: ${\displaystyle \textstyle \rho _{1}\oplus \rho _{2}}$ ${\displaystyle \textstyle \mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /3\mathbb {Z} }$ ${\displaystyle \textstyle \mathbb {C} ^{2}\oplus \mathbb {C} ^{3}=\mathbb {C} ^{5},}$ ${\displaystyle k\in \mathbb {Z} /2\mathbb {Z} ,l\in \mathbb {Z} /3\mathbb {Z} }$

${\displaystyle \rho _{1}\oplus \rho _{2}(k,l)=\left({\begin{array}{cc}\rho _{1}(k)&0\\0&\rho _{2}(l)\end{array}}\right).}$

Since it is sufficient to give the picture of the group's creator, we note that is given by: ${\displaystyle \textstyle \rho _{1}\oplus \rho _{2}\colon \mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /3\mathbb {Z} \to {\text{GL}}_{5}(\mathbb {C} )}$

${\displaystyle \rho _{1}\oplus \rho _{2}({\overline {1}},{\overline {1}})=\left({\begin{array}{ccccc}0&-i&0&0&0\\i&0&0&0&0\\0&0&1&0&e^{\frac {2\pi i}{3}}\\0&0&0&e^{\frac {2\pi i}{3}}&0\\0&0&0&0&e^{\frac {4\pi i}{3}}\end{array}}\right).}$

See also

• Direct sum (Banach space)
• Direct sum (Abelian group)
• Orthogonal sum , extension to spaces provided with scalar products , especially Hilbert spaces .

literature

• Gerd Fischer: Lineare Algebra , Vieweg-Verlag, ISBN 3-528-03217-0