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Kronecker Sum


The Kronecker sum is the matrix sum defined by

 A direct sum B=A tensor I_b+I_a tensor B,
(1)

where A and B are square matrices of order a and b, respectively, I_n is the identity matrix of order n, and  tensor denotes the Kronecker product.

For example, the Kronecker sum of two 2×2 matrices (a)_(ij) and (b)_(ij) 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)].
(2)

The Kronecker sum satisfies the nice property

 exp(A) tensor exp(B)=exp(A direct sum B),
(3)

where exp(A) denotes a matrix exponential.


See also

Kronecker Product, Matrix Direct Sum

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References

Horn, R. A. and Johnson, C. R. Topics in Matrix Analysis. Cambridge, England: Cambridge University Press, p. 208, 1994.

Referenced on Wolfram|Alpha

Kronecker Sum

Cite this as:

Weisstein, Eric W. "Kronecker Sum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KroneckerSum.html

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