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The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebra (PHA) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product(STP) of matrices are reviewed. The zero set is defined for non-invertible hypercomplex numbers in a given PHA, and a characteristic function is proposed for calculating zero set. Then PHA of different dimensions are considered. First, $2$-dimensional PHAs are considered as examples to calculate their zero sets etc. Second, all the $3$-dimensional PHAs are obtained and the corresponding zero sets are investigated. Third, $4$-dimensional or even higher dimensional PHAs are also considered. Finally, matrices over pre-assigned PHA, called perfect hypercomplex matrices (PHMs) are considered. Their properties are also investigated.
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This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described, and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode.
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