Can Real Matrices Be Transformed into Complex Matrices Through Similarity?

[sweetiepi]

Let A and B be 2 x 2 real matrices such that A = PBP^-1 for some invertible 2 x 2 complex matrix P. Prove that A = QBQ^-1 for some invertible 2 x 2 real matrix Q.

 

[Eynstone]

A plump calculation shows that Q = PP* ( 8 denotes complex conjugation here).