JORDAN CANONICAL FORM OF A PARTITIONED COMPLEX MATRIX AND ITS APPLICATION TO REAL QUATERNION MATRICES

Let ∑ be the collection of all 2n × 2n partitioned complex matrices (A ₁/-A ₂ A ₂/A ₁), where A ₁ and A ₂ 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.