Bulletin of the
Korean Mathematical Society BKMS

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 $T^{*}$.

Keywords: Irreducible operator, unitary similarity, Kronecker sum, symmetric tensor

MSC numbers: Primary 47A15, 15A21; Secondary 47L40, 15A69

Supported by: We thank Bill and Linda Frost Fund for the Frost Student Research Award to fund this project. Jaehui Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (Grant No. NRF-2018R1A2B6004116).