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
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