HOW TO DISTINGUISH BETWEEN THE LATENTLY REAL MATRICES AND THE BLOCK QUATERNIONS?

摘要

Let a complex n × n matrix A be unitarily similar to its entry wise conjugate A. If the unitary matrix P in the relation A = P~* AP can be chosen symmetric (skew-symmetric), then A is called a latently real matrix (respectively, a generalized block quaternion). Only these two cases are possible if A is a (unitarily) irreducible matrix. The following question is discussed: How to find out whether a given A is latently real matrix or a generalized block quaternion.