# How can I compute a Kronecker sum in Mathematica?

There is Kronecker product but there is no Kronecker sum? It seems like a very important features to include.

So in the absence of a Kronecker sum function, how can I construct my own Kronecker sum $A\oplus B$ of two arbitrary $n\times n$ matrices $A$ and $B$?

Thanks very much!

• Your question seems ambiguous. The matrix direct sum and the Kronecker sum are two different things. Which one do you want? – MarcoB Apr 25 '18 at 17:44
• @MarcoB you're absolutely right, I mean the Kronecker sum – Turbotanten Apr 25 '18 at 17:52
• Thank you for clarifying. You got both answers at this point anyway :-) , but it may still be best to clear that up in the text of the question. – MarcoB Apr 25 '18 at 17:54
• @MarcoB That's great because I actually needed both. – Turbotanten Apr 25 '18 at 17:57
• See also Henrik's answer here. I didn't realize I was trying to get the Kronecker sum, but he wrote an amazing compiled implementation for sparse matrices. – b3m2a1 Jul 27 '18 at 21:32

Using the Wikipedia definition of Kronecker sum, it seems that we can define it in terms of the Kronecker products, which is built in:

Clear[kroneckersum]
kroneckersum[a_, b_ /; Dimensions[a] == Dimensions[b]] :=
KroneckerProduct[a, IdentityMatrix[Length[a]]] +
KroneckerProduct[IdentityMatrix[Length[b]], b]

a = RandomInteger[{0, 10}, {5, 5}]
b = RandomInteger[{0, 10}, {5, 5}]

kroneckersum[a, b]


An alternative implementation that has the significant advantage of retaining the use of SparseArrays for large matrices was proposed by Henrik in comments:

kroneckersum[a_?SquareMatrixQ, b_?SquareMatrixQ] :=
KroneckerProduct[a, IdentityMatrix[Length[b], SparseArray]] +
KroneckerProduct[IdentityMatrix[Length[a], SparseArray], b]


This also reminded me of SquareMatrixQ, a convenient bit of syntactic sugar which I'd seen used before, but keep forgetting.

• I think there is no reason for checking dimensions if you use kroneckersum[a_?SquareMatrixQ, b_?SquareMatrixQ] := KroneckerProduct[a, IdentityMatrix[Length[b], SparseArray]] + KroneckerProduct[IdentityMatrix[Length[a], SparseArray], b] . Due to using SparseArrays, this works also for larger matrices (e.g., I tested a = RandomInteger[{0, 10}, {250, 250}]; b = RandomInteger[{0, 10}, {350, 350}];. – Henrik Schumacher Apr 25 '18 at 18:21
• @HenrikSchumacher That's very nice! I particularly like the SparseArray part. Would you mind if I added / edited your method in the answer? – MarcoB Apr 25 '18 at 20:06
• Of course I don't mind. That was the purpose of my comment. – Henrik Schumacher Apr 25 '18 at 20:22
• @HenrikSchumacher Done. Thank you. – MarcoB Apr 25 '18 at 20:27

You could use DiagonalMatrix and ArrayFlatten to define a direct sum:

DirectSum[a_List] := ArrayFlatten @ Block[{Identity}, DiagonalMatrix[Identity/@a]]


For instance:

DirectSum[{
{{a,b,c},{d,e,f}},
{{g,h},{i,j},{k,l}}
}] //TeXForm


$\left( \begin{array}{ccccc} a & b & c & 0 & 0 \\ d & e & f & 0 & 0 \\ 0 & 0 & 0 & g & h \\ 0 & 0 & 0 & i & j \\ 0 & 0 & 0 & k & l \\ \end{array} \right)$

See @MarcoB's answer if you wanted the Kronecker sum.

From the docs for KroneckerProduct:

KroneckerSum[a_, b_] /; MatrixQ[a] && MatrixQ[b] :=
Catch@Module[{n, p, m, q},
{n, p} = Dimensions[a]; {m, q} = Dimensions[b];
If[n != p || m != q, Throw[\$Failed]];
KroneckerProduct[a, IdentityMatrix[m]] +
KroneckerProduct[IdentityMatrix[n], b]
];