The Kronecker sum is the matrix sum defined by
 |
(1)
|
where 
and 
are square
matrices of order 
and 
, respectively, 
is the identity matrix
of order 
,
and 
denotes the Kronecker
product.
For example, the Kronecker sum of two 
matrices 
and 
is given by
![[a_(11) a_(12); a_(21) a_(22)] direct sum [b_(11) b_(12); b_(21) b_(22)]
=[a_(11)+b_(11) b_(12) a_(12) 0; b_(21) a_(11)+b_(22) 0 a_(12); a_(21) 0 a_(22)+b_(11) b_(12); 0 a_(21) b_(21) a_(22)+b_(22)].](/images/equations/KroneckerSum/NumberedEquation2.svg) |
(2)
|
The Kronecker sum satisfies the nice property
 |
(3)
|
where 
denotes a matrix exponential.
