Involutions and diagonalizability

Hi,

Here's a problem I wasn't able to solve last quarter:
"Let A and B be two commuting nxn real matrices with A^2=B^2=I. Prove
A and B are simultaneously diagonalizable."

I can't even figure out how to prove A is diagonalizable. If we
consider it as complex, it's similar to something upper triangular, and
it's easy to see the diagonals in the upper triangular similar matrix
are all plus or minus 1, but I can't figure out how to get
diagonalizable. If I could show A and B alone were diagonalizable, I
think I have a good idea how to finish the problem.

.

Relevant Pages

  • Re: Involutions and diagonalizability
    ... it's easy to see the diagonals in the upper triangular similar matrix ... What happens when you take the square of a matrix ...
    (sci.math)
  • Re: Involutions and diagonalizability
    ... it's easy to see the diagonals in the upper triangular similar matrix ... polynomial m_Mhas only simple zeros over the algebraic closure of k. ... Then note that a regular complex matrix M is diagonalizable if M^n is ...
    (sci.math)
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