Maths - Kronecker Sum

Notation

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:

H =³

Which says that quaternions are made up of a one dimensional scalar combined with a 3 dimensional vector like (actually bivector) quantity. Or:

H =CCj

Which says that quaternions are made up of two complex number algebras, or:

G3+ = ^²³

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 as instead 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' )

where:

• × 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 *.

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 R^m and Rⁿ is R^{m + n}.

If we are multiplying two integers n and m then we could say,

	n
n*m =	∑ m
	1=1

In the same way if we are raising n to the m^th power then we could say,

	n
nm =	∏ m
	1=1

A direct product for a finite index is identical to the direct sum:

n		n
	=	∏
i=1		1=1

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.

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 = (I_mA) + (BI_n).

where:

• AR_n×n
• BR_m×m

Example

In order to illustrate now to calculate the Kronecker sum here is an example:

If A =	3 4 9 -4 3 6 3 -7 8
and B =	5 -3 2 4

calculate AB

So in this case A is 3×3 so In=	1 0 0 0 1 0 0 0 1
and B is 2 ×2 so Im=	1 0 0 1

So AB = (I_mA) + (BI_n) gives:

AB=

3 4 9 0 0 0 -4 3 6 0 0 0 3 -7 8 0 0 0 0 0 0 3 4 9 0 0 0 -4 3 6 0 0 0 3 -7 8

+

5 0 0 -3 0 0 0 5 0 0 -3 0 0 0 5 0 0 -3 2 0 0 4 0 0 0 2 0 0 4 0 0 0 2 0 0 4

which gives

AB=	8 4 9 -3 0 0 -4 8 6 0 -3 0 3 -7 13 0 0 -3 2 0 0 7 4 9 0 2 0 -4 7 6 0 0 2 3 -7 12

General Case

Given K square matrices: A_K,…,A₁, where A_l is of size n_l×n_l,

their Kronecker sum is

K≥l≥1

Al=

∑ K≥l≥1

InK…nl+1AlInl−1…n1

∈ R^{nK…n1×nK…n1}

where:

• Im = identity matrix of size m × m
• A[i,j] = A_K[i_K,j_K] · A_K−1[i_K−1,j_K−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

nonzeros:

η ( ⊕K≥l≥1Al) ≤∑K≥l≥1η(Al)nl· ∏K≥h≥1

Properties

The Kronecker sum commutes:

AB = BA.

Group Theory