Can Real Matrices Be Transformed into Complex Matrices Through Similarity?

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SUMMARY

The discussion centers on the transformation of real matrices into complex matrices through similarity. It establishes that for 2 x 2 real matrices A and B, if A can be expressed as A = PBP^-1 using an invertible complex matrix P, then A can also be represented as A = QBQ^-1 for some invertible 2 x 2 real matrix Q. The calculation provided indicates that Q is derived from the product of P and its complex conjugate, denoted as Q = PP*.

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

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