Thread: 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!

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

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

For the second, consider the 90 degree rotation matrix $\displaystyle \begin{bmatrix} 0 & -1
\\ 1 & 0 \end{bmatrix}$