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

Let Sigma be the collection of all 2n x 2n partitioned complex matrices (A(1)/-A(2) A(2)/A(1)), where A(1) and A(2) are n x n complex matrices, the bars on top of them mea n matrix conjugate. We show that Sigma is closed under similarity transform ation to Jordan (canonical) forms. Precisely, any matrix in Sigma is simila r to a matrix in the form J circle plus (J) over bar is an element of Sigma via an invertible matrix in Sigma, where J is a Jordan form whose diagonal elements all have nonnegative imaginary parts. An application of this resu lt gives the Jordan form of real quaternion matrices.