Let A and B be 2x2 real matricies, and suppose there exists an invertible complex 2x2 matrix P such that B = [P^(-1)]AP.(adsbygoogle = window.adsbygoogle || []).push({});

Show that there exists a real invertible 2x2 matrix Q such that B = [Q^(-1)]AQ.

A and B are similar when thought of as complex matricies, so they represent the same linear transformation on C2 for appropriately chosen bases, and share many other properties:

same trace, same determinant, same characteristic equation , same eigenvalues.

If I take Q = (1/2)(P + P bar) (the "real part" of P),

then I can show QB = AQ, and so B = [Q^(-1)]AQ if Q is invertible, but this Q may not be invertible.

I also noticed that each of A and B may be triangularized (since each of A and B has an eigenvalue), but I don't know where to go from there...

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