Jordan canonical form of a partitioned complex matrix and its application to real quaternion matrices

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Communications in Algebra, ISSN: 0092-7872, Vol: 29, Issue: 6, Page: 2363-2375

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Zhang, Fuzhen; Wei, Yimin
Informa UK Limited; NSUWorks
article description
Let ∑ be the collection of all 2n × 2n partitioned complex matrices (A/-A A/A), where A and A are n × n complex matrices, the bars on top of them mean matrix conjugate. We show that ∑ is closed under similarity transformation to Jordan (canonical) forms. Precisely, any matrix in ∑ is similar to a matrix in the form J ⊗ J̄ ∈ ∑ via an invertible matrix in ∑, where J is a Jordan form whose diagonal elements all have nonnegative imaginary parts. An application of this result gives the Jordan form of real quaternion matrices.