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Decomposition of the Kronecker sums of matrices into a direct sum of irreducible matrices

박재휘|Caixing Gu|Chase Peak|Jordan Rowley
대한수학회 (The Korean Mathematical Society)|2021.05|대한수학회보 (Bulletin of the KMS)|제58권 제3호|pp. 637-657|DOI 10.4134/BKMS.b200437

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키워드

Irreducible operatorunitary similarityKronecker sumsymmetric tensor
주제분류수학

초록

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 $T^{*}$.

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 $T^{*}$.

참고문헌 (33)

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