0
$\begingroup$

On two occasions I have come upon problems requiring the differences between each point in two matrices or vectors. One such example is to find the extremum points in two arbitrary curves. The solution was seen in the Kronecker product $(\mathbf M=\mathbf A\otimes \mathbf B)$, albeit with a summation rather than a product. I happily called this the Kronecker sum, although I've seen it referred to as the Kronecker product w.r.t. addition.

Now it comes to my attention that my chosen terminology is not only taken, but it has a different meaning. See, for example, Weisstein, Eric W. "Kronecker Sum." From MathWorld--A Wolfram Web Resource. This Kronecker sum is given by $(\mathbf M=\mathbf A\oplus \mathbf B)$.

My question is, does the Kronecker product w.r.t. addition have is own name and symbol?

$\endgroup$
  • $\begingroup$ MathWorld has many errors. Wikipedia has a definition of Kronecker sum which maybe is the operation you want, although I'm not sure I've correctly parsed your description of it: en.wikipedia.org/wiki/… $\endgroup$ – Qiaochu Yuan Mar 7 at 20:16
  • $\begingroup$ @QiaochuYuan The definition in your link is the same as that of MathWorld. Notice that it applies only to square matrices. However, let me add, that my matrices are actually a column vector and a row vector. When I tested my results I found that indeed $e^{A\oplus B}=e^A \otimes} e^B$, as it should. So perhaps MathWorld and Wikipedia give definitions that are too stringent. Nevertheless, in the final analysis, my algorithm is more direct, and faster, as well. $\endgroup$ – Cye Waldman Mar 7 at 20:44
  • $\begingroup$ Oh, my bad for not actually following the link. That operation really should not be written $\oplus$; that should refer to the direct sum. In that case I have no idea what operation you're referring to. What do you mean by $e^A$ if $A$ isn't a square matrix? $\endgroup$ – Qiaochu Yuan Mar 7 at 20:46
  • $\begingroup$ $e^A$ is just the exponential of the elements of the vector. So I'm not sure I understand your question. PS, forgive the messy MathJax in my first reply, I'm trying to replicate the identity in Wikipedia. $\endgroup$ – Cye Waldman Mar 7 at 20:51
  • $\begingroup$ That is not what either MathWorld or Wikipedia mean by exponential. They mean the matrix exponential, which only operates on square matrices: en.wikipedia.org/wiki/Matrix_exponential I continue to have no idea what operation you're talking about. Can you just write out a formula for it? $\endgroup$ – Qiaochu Yuan Mar 7 at 20:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.