Jordan Canonical Form of a Partitioned Complex Matrix and Its Application to Real Quaternion Matrices

Let Σ be the collection of all 2n × 2n partitioned complex matrices where A 1 and A 2 are n × n complex matrices, the bars on top of them mean matrix conjugate. We show that Σ is closed under similarity transformation to Jordan (canonical) forms. Precisely, any matrix in Σ is similar to a matrix in the form J ⊕ ∈ Σ via an invertible matrix in Σ, where J is a Jordan form whose diagonalelements all have nonnegative imaginary parts. An application of this result gives the Jordan form of real quaternion matrices.