The symbolseems to be used to a number of related concepts:
- The Kronecker sum or tensor sum
- The direct sum
- Group direct product
- Direct product of modules
The direct sum
A construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as subspaces.
As far as I can tell, this does not presicely define the Cayley table, for example we could say:
Which says that quaternions are made up of a one dimensional scalar combined with a 3 dimensional vector like (actually bivector) quantity. Or:
Which says that quaternions are made up of two complex number algebras, or:
Which says that the even geometric algebra based on 3 dimensions which square to +ve is made up of a one dimensional scalar combined with a 3 dimensional bivector quantity.
Group direct product
For abelian groups (groups which commute) we can represent the group operation asinstead of the usual multipication, in this case the identity element will be '0' and the inverse operation will be -x.
We can also use this to combine algebras, for instance if we have the algebras G and H with sample elements g and h. Combining these algebras can be defined elementwise:
(g, h) × (g' , h' ) = (g * g' , h o h' )
- × is the operation of the combined algebra.
- * is the operation of the group G.
- o is the operation of the group H which may be, or maynot be, the same as *.
For more information about combining groups using external or internal products see this page.
Direct product of modules
The direct product for modules (not to be confused with the tensor product) is very similar to the one defined for groups above, using the cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. Starting from R we get Euclidean space Rn, the prototypical example of a real n-dimensional vector space. The direct product of Rm and Rn is Rm + n.
If we are multiplying two integers n and m then we could say,
|n*m =||∑ m|
In the same way if we are raising n to the mth power then we could say,
|nm =||∏ m|
A direct product for a finite index is identical to the direct sum:
The direct sum and direct product differ only for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual: the direct sum is the coproduct, while the direct product is the product.
Cartesian and Tensor Products in other Algebras
|Cartesian||Cartesian product||Kronecker sum|
|Tensor||tensor product||Kronecker product|
Cartesian and tensor products not only occur in matrix algebra but also occur in other algebras. It may help intuative understanding to look at these products in graph theory on the page here.
Other meanings of
The symbolis also used for the binary 'exclusive or' operation although
Kronecker Sum of matrix with real terms
The Kronecker sum (or tensor sum) of A and B, denoted AB, is the mn×mn matrix:
AB = (ImA) + (BIn).
In order to illustrate now to calculate the Kronecker sum here is an example:
|If A =||
|and B =||
|So in this case A is 3×3 so In=||
|and B is 2 ×2 so Im=||
So AB = (ImA) + (BIn) gives:
Given K square matrices: AK,…,A1, where Al is of size nl×nl,
their Kronecker sum is
- Im = identity matrix of size m × m
- A[i,j] = AK[iK,jK] · AK−1[iK−1,jK−1]…A1[i1,j1]
using the mixed-base numbering scheme (indices start at 0)
i = (...((iK) · nK−1+ iK−1) · nK−2…) · n1+ i1= ∑K≥l≥1il· ∏l>h≥1nh
η ( ⊕K≥l≥1Al) ≤∑K≥l≥1η(Al)nl· ∏K≥h≥1
The Kronecker sum commutes:
AB = BA.