Quasi-real normal matrices and eigenvalue pairings

For square complex matrices A and B of the same size, commutativity-like relations such as AB=±BA, AB=±BA*, AB=±BAT, AB=±BTA, etc., often cause a special structure of A to be reflected in some special structure for B. We study eigenvalue pairing theorems for B when A is quasi-real normal (QRN), a class of complex matrices that is a natural generalization of the real normal matrices. A new canonical form for QRN matrices is an essential tool in our development.