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 Decomposition of the Kronecker sums of matrices into a direct sum of irreducible matrices Bull. Korean Math. Soc.Published online February 24, 2021 Caixing Gu, Jaehui Park, Chase Peak, and Jordan Rowley California Polytechnic State University, Seoul National University Abstract : In this paper we decompose (under unitary similarity) the Kronecker sum $A\boxplus A$ ($=A\otimes I+I\otimes A$) into a direct sum of irreducible matrices, when $A$ is a $3\times3$ matrix. As a consequence we identify $\mathcal{K}(A\boxplus A)$ as the direct sum of several full matrix algebras as predicted by Artin--Wedderburn theorem, where $\mathcal{K}(T)$ is the unital algebra generated by $T$ and $T^{\ast}$. Keywords : irreducible operators, unitary similarity, Kronecker sum, symmetric tensor MSC numbers : 47A15, 15A21 Full-Text :