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

Abstract

Let ∑ be the collection of all 2n × 2n partitioned complex matrices (A1/-A2 A2/A1), where A1 and A2 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.