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Math Help - diagonalizability of B if AB=BA

  1. #1
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    diagonalizability of B if AB=BA

    If A is diagonalizable and AB=BA, then is B diagonalizable? If so, how would I go about proving this? I think that this is true, but I am not positive.

    Also, is there a real 2x2 matrix A satisfying A squared=-I? I can't think of one off of the top of my head.

    Thanks!
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  2. #2
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    If A is diagonalizable, then there exists an invertible matrix P and a diagonal matrix D such that A = PDP^{-1}. Specifically, A is also invertible since det(A) = det(PDP^{-1}) = det(P)det(D)det(P^{-1}) = det(P)\cdot \frac{1}{det(P)}\cdot det(D) = det(D) but D is diagonal and therefore obviously det(D) \neq 0 \Rightarrow det(A) \neq 0.

    Now, what does this tell you about the original equation?

    For the second, consider the 90 degree rotation matrix \begin{bmatrix} 0 & -1 <br />
\\ 1 & 0 \end{bmatrix}
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