Decomposition of the Kronecker sums of matrices into a direct sum of irreducible matrices
Bull. Korean Math. Soc. 2021 Vol. 58, No. 3, 637-657
Published online February 24, 2021
Printed May 31, 2021
Caixing Gu, Jaehui Park, Chase Peak, Jordan Rowley
California Polytechnic State University; Seoul National University; California Polytechnic State University; California Polytechnic State 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 $T^{*}$.
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).
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