diagonalizability of B if AB=BA

**ecc5** · Dec 10th 2009, 11:41 AM

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!

**Defunkt** · Dec 10th 2009, 12:18 PM

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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